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G = C7×C22.57C24  order 448 = 26·7

Direct product of C7 and C22.57C24

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Aliases: C7×C22.57C24, C14.1722+ 1+4, C14.1242- 1+4, C4⋊Q8⋊18C14, C22⋊Q8⋊20C14, C42⋊2C2⋊9C14, C42.55(C2×C14), C42.C2⋊12C14, C4.4D4.9C14, (C2×C28).684C23, (C2×C14).383C24, (C4×C28).296C22, (D4×C14).225C22, C23.26(C22×C14), C22.57(C23×C14), (Q8×C14).188C22, C22.D4.3C14, C2.16(C7×2- 1+4), C2.24(C7×2+ 1+4), (C22×C28).463C22, (C22×C14).109C23, (C7×C4⋊Q8)⋊39C2, C4⋊C4.35(C2×C14), (C7×C22⋊Q8)⋊47C2, (C2×D4).38(C2×C14), C22⋊C4.8(C2×C14), (C2×Q8).31(C2×C14), (C7×C42.C2)⋊29C2, (C7×C42⋊2C2)⋊20C2, (C7×C4⋊C4).252C22, (C22×C4).74(C2×C14), (C2×C4).43(C22×C14), (C7×C4.4D4).18C2, (C7×C22⋊C4).93C22, (C7×C22.D4).6C2, SmallGroup(448,1346)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C22 — C7×C22.57C24
Chief series
C1 — C2 — C22 — C2×C14 — C2×C28 — Q8×C14 — C7×C4⋊Q8 — C7×C22.57C24
Lower central
C1 — C22 — C7×C22.57C24
Upper central
C1 — C2×C14 — C7×C22.57C24

Generators and relations for C7×C22.57C24
 G = < a,b,c,d,e,f,g | a7=b2=c2=g2=1, d2=e2=f2=b, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, ede-1=bd=db, geg=be=eb, bf=fb, bg=gb, fdf-1=cd=dc, ce=ec, cf=fc, cg=gc, gdg=bcd, fef-1=bce, fg=gf >

Subgroups: 282 in 196 conjugacy classes, 142 normal (18 characteristic)
C1, C2, C2, C2, C4, C22, C22, C7, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C14, C14, C14, C42, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C2×Q8, C28, C2×C14, C2×C14, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C42⋊2C2, C4⋊Q8, C2×C28, C2×C28, C2×C28, C7×D4, C7×Q8, C22×C14, C22.57C24, C4×C28, C4×C28, C7×C22⋊C4, C7×C4⋊C4, C22×C28, D4×C14, Q8×C14, Q8×C14, C7×C22⋊Q8, C7×C22.D4, C7×C4.4D4, C7×C42.C2, C7×C42⋊2C2, C7×C4⋊Q8, C7×C22.57C24
Quotients: C1, C2, C22, C7, C23, C14, C24, C2×C14, 2+ 1+4, 2- 1+4, C22×C14, C22.57C24, C23×C14, C7×2+ 1+4, C7×2- 1+4, C7×C22.57C24

Smallest permutation representation of C7×C22.57C24
►On 224 points
Generators in S224
(1 2 3 4 5 6 7)(8 9 10 11 12 13 14)(15 16 17 18 19 20 21)(22 23 24 25 26 27 28)(29 30 31 32 33 34 35)(36 37 38 39 40 41 42)(43 44 45 46 47 48 49)(50 51 52 53 54 55 56)(57 58 59 60 61 62 63)(64 65 66 67 68 69 70)(71 72 73 74 75 76 77)(78 79 80 81 82 83 84)(85 86 87 88 89 90 91)(92 93 94 95 96 97 98)(99 100 101 102 103 104 105)(106 107 108 109 110 111 112)(113 114 115 116 117 118 119)(120 121 122 123 124 125 126)(127 128 129 130 131 132 133)(134 135 136 137 138 139 140)(141 142 143 144 145 146 147)(148 149 150 151 152 153 154)(155 156 157 158 159 160 161)(162 163 164 165 166 167 168)(169 170 171 172 173 174 175)(176 177 178 179 180 181 182)(183 184 185 186 187 188 189)(190 191 192 193 194 195 196)(197 198 199 200 201 202 203)(204 205 206 207 208 209 210)(211 212 213 214 215 216 217)(218 219 220 221 222 223 224)

(1 34)(2 35)(3 29)(4 30)(5 31)(6 32)(7 33)(8 219)(9 220)(10 221)(11 222)(12 223)(13 224)(14 218)(15 22)(16 23)(17 24)(18 25)(19 26)(20 27)(21 28)(36 46)(37 47)(38 48)(39 49)(40 43)(41 44)(42 45)(50 74)(51 75)(52 76)(53 77)(54 71)(55 72)(56 73)(57 64)(58 65)(59 66)(60 67)(61 68)(62 69)(63 70)(78 102)(79 103)(80 104)(81 105)(82 99)(83 100)(84 101)(85 92)(86 93)(87 94)(88 95)(89 96)(90 97)(91 98)(106 130)(107 131)(108 132)(109 133)(110 127)(111 128)(112 129)(113 120)(114 121)(115 122)(116 123)(117 124)(118 125)(119 126)(134 158)(135 159)(136 160)(137 161)(138 155)(139 156)(140 157)(141 148)(142 149)(143 150)(144 151)(145 152)(146 153)(147 154)(162 186)(163 187)(164 188)(165 189)(166 183)(167 184)(168 185)(169 176)(170 177)(171 178)(172 179)(173 180)(174 181)(175 182)(190 214)(191 215)(192 216)(193 217)(194 211)(195 212)(196 213)(197 204)(198 205)(199 206)(200 207)(201 208)(202 209)(203 210)

(1 45)(2 46)(3 47)(4 48)(5 49)(6 43)(7 44)(8 18)(9 19)(10 20)(11 21)(12 15)(13 16)(14 17)(22 223)(23 224)(24 218)(25 219)(26 220)(27 221)(28 222)(29 37)(30 38)(31 39)(32 40)(33 41)(34 42)(35 36)(50 70)(51 64)(52 65)(53 66)(54 67)(55 68)(56 69)(57 75)(58 76)(59 77)(60 71)(61 72)(62 73)(63 74)(78 98)(79 92)(80 93)(81 94)(82 95)(83 96)(84 97)(85 103)(86 104)(87 105)(88 99)(89 100)(90 101)(91 102)(106 126)(107 120)(108 121)(109 122)(110 123)(111 124)(112 125)(113 131)(114 132)(115 133)(116 127)(117 128)(118 129)(119 130)(134 154)(135 148)(136 149)(137 150)(138 151)(139 152)(140 153)(141 159)(142 160)(143 161)(144 155)(145 156)(146 157)(147 158)(162 182)(163 176)(164 177)(165 178)(166 179)(167 180)(168 181)(169 187)(170 188)(171 189)(172 183)(173 184)(174 185)(175 186)(190 210)(191 204)(192 205)(193 206)(194 207)(195 208)(196 209)(197 215)(198 216)(199 217)(200 211)(201 212)(202 213)(203 214)

(1 134 34 158)(2 135 35 159)(3 136 29 160)(4 137 30 161)(5 138 31 155)(6 139 32 156)(7 140 33 157)(8 131 219 107)(9 132 220 108)(10 133 221 109)(11 127 222 110)(12 128 223 111)(13 129 224 112)(14 130 218 106)(15 117 22 124)(16 118 23 125)(17 119 24 126)(18 113 25 120)(19 114 26 121)(20 115 27 122)(21 116 28 123)(36 141 46 148)(37 142 47 149)(38 143 48 150)(39 144 49 151)(40 145 43 152)(41 146 44 153)(42 147 45 154)(50 186 74 162)(51 187 75 163)(52 188 76 164)(53 189 77 165)(54 183 71 166)(55 184 72 167)(56 185 73 168)(57 176 64 169)(58 177 65 170)(59 178 66 171)(60 179 67 172)(61 180 68 173)(62 181 69 174)(63 182 70 175)(78 214 102 190)(79 215 103 191)(80 216 104 192)(81 217 105 193)(82 211 99 194)(83 212 100 195)(84 213 101 196)(85 204 92 197)(86 205 93 198)(87 206 94 199)(88 207 95 200)(89 208 96 201)(90 209 97 202)(91 210 98 203)

(1 78 34 102)(2 79 35 103)(3 80 29 104)(4 81 30 105)(5 82 31 99)(6 83 32 100)(7 84 33 101)(8 163 219 187)(9 164 220 188)(10 165 221 189)(11 166 222 183)(12 167 223 184)(13 168 224 185)(14 162 218 186)(15 180 22 173)(16 181 23 174)(17 182 24 175)(18 176 25 169)(19 177 26 170)(20 178 27 171)(21 179 28 172)(36 85 46 92)(37 86 47 93)(38 87 48 94)(39 88 49 95)(40 89 43 96)(41 90 44 97)(42 91 45 98)(50 130 74 106)(51 131 75 107)(52 132 76 108)(53 133 77 109)(54 127 71 110)(55 128 72 111)(56 129 73 112)(57 120 64 113)(58 121 65 114)(59 122 66 115)(60 123 67 116)(61 124 68 117)(62 125 69 118)(63 126 70 119)(134 190 158 214)(135 191 159 215)(136 192 160 216)(137 193 161 217)(138 194 155 211)(139 195 156 212)(140 196 157 213)(141 197 148 204)(142 198 149 205)(143 199 150 206)(144 200 151 207)(145 201 152 208)(146 202 153 209)(147 203 154 210)

(1 50 34 74)(2 51 35 75)(3 52 29 76)(4 53 30 77)(5 54 31 71)(6 55 32 72)(7 56 33 73)(8 191 219 215)(9 192 220 216)(10 193 221 217)(11 194 222 211)(12 195 223 212)(13 196 224 213)(14 190 218 214)(15 208 22 201)(16 209 23 202)(17 210 24 203)(18 204 25 197)(19 205 26 198)(20 206 27 199)(21 207 28 200)(36 57 46 64)(37 58 47 65)(38 59 48 66)(39 60 49 67)(40 61 43 68)(41 62 44 69)(42 63 45 70)(78 126 102 119)(79 120 103 113)(80 121 104 114)(81 122 105 115)(82 123 99 116)(83 124 100 117)(84 125 101 118)(85 131 92 107)(86 132 93 108)(87 133 94 109)(88 127 95 110)(89 128 96 111)(90 129 97 112)(91 130 98 106)(134 175 158 182)(135 169 159 176)(136 170 160 177)(137 171 161 178)(138 172 155 179)(139 173 156 180)(140 174 157 181)(141 163 148 187)(142 164 149 188)(143 165 150 189)(144 166 151 183)(145 167 152 184)(146 168 153 185)(147 162 154 186)

(8 18)(9 19)(10 20)(11 21)(12 15)(13 16)(14 17)(22 223)(23 224)(24 218)(25 219)(26 220)(27 221)(28 222)(78 102)(79 103)(80 104)(81 105)(82 99)(83 100)(84 101)(85 92)(86 93)(87 94)(88 95)(89 96)(90 97)(91 98)(106 130)(107 131)(108 132)(109 133)(110 127)(111 128)(112 129)(113 120)(114 121)(115 122)(116 123)(117 124)(118 125)(119 126)(134 147)(135 141)(136 142)(137 143)(138 144)(139 145)(140 146)(148 159)(149 160)(150 161)(151 155)(152 156)(153 157)(154 158)(162 175)(163 169)(164 170)(165 171)(166 172)(167 173)(168 174)(176 187)(177 188)(178 189)(179 183)(180 184)(181 185)(182 186)(190 210)(191 204)(192 205)(193 206)(194 207)(195 208)(196 209)(197 215)(198 216)(199 217)(200 211)(201 212)(202 213)(203 214)
 
G:=sub<Sym(224)| (1,2,3,4,5,6,7)(8,9,10,11,12,13,14)(15,16,17,18,19,20,21)(22,23,24,25,26,27,28)(29,30,31,32,33,34,35)(36,37,38,39,40,41,42)(43,44,45,46,47,48,49)(50,51,52,53,54,55,56)(57,58,59,60,61,62,63)(64,65,66,67,68,69,70)(71,72,73,74,75,76,77)(78,79,80,81,82,83,84)(85,86,87,88,89,90,91)(92,93,94,95,96,97,98)(99,100,101,102,103,104,105)(106,107,108,109,110,111,112)(113,114,115,116,117,118,119)(120,121,122,123,124,125,126)(127,128,129,130,131,132,133)(134,135,136,137,138,139,140)(141,142,143,144,145,146,147)(148,149,150,151,152,153,154)(155,156,157,158,159,160,161)(162,163,164,165,166,167,168)(169,170,171,172,173,174,175)(176,177,178,179,180,181,182)(183,184,185,186,187,188,189)(190,191,192,193,194,195,196)(197,198,199,200,201,202,203)(204,205,206,207,208,209,210)(211,212,213,214,215,216,217)(218,219,220,221,222,223,224), (1,34)(2,35)(3,29)(4,30)(5,31)(6,32)(7,33)(8,219)(9,220)(10,221)(11,222)(12,223)(13,224)(14,218)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28)(36,46)(37,47)(38,48)(39,49)(40,43)(41,44)(42,45)(50,74)(51,75)(52,76)(53,77)(54,71)(55,72)(56,73)(57,64)(58,65)(59,66)(60,67)(61,68)(62,69)(63,70)(78,102)(79,103)(80,104)(81,105)(82,99)(83,100)(84,101)(85,92)(86,93)(87,94)(88,95)(89,96)(90,97)(91,98)(106,130)(107,131)(108,132)(109,133)(110,127)(111,128)(112,129)(113,120)(114,121)(115,122)(116,123)(117,124)(118,125)(119,126)(134,158)(135,159)(136,160)(137,161)(138,155)(139,156)(140,157)(141,148)(142,149)(143,150)(144,151)(145,152)(146,153)(147,154)(162,186)(163,187)(164,188)(165,189)(166,183)(167,184)(168,185)(169,176)(170,177)(171,178)(172,179)(173,180)(174,181)(175,182)(190,214)(191,215)(192,216)(193,217)(194,211)(195,212)(196,213)(197,204)(198,205)(199,206)(200,207)(201,208)(202,209)(203,210), (1,45)(2,46)(3,47)(4,48)(5,49)(6,43)(7,44)(8,18)(9,19)(10,20)(11,21)(12,15)(13,16)(14,17)(22,223)(23,224)(24,218)(25,219)(26,220)(27,221)(28,222)(29,37)(30,38)(31,39)(32,40)(33,41)(34,42)(35,36)(50,70)(51,64)(52,65)(53,66)(54,67)(55,68)(56,69)(57,75)(58,76)(59,77)(60,71)(61,72)(62,73)(63,74)(78,98)(79,92)(80,93)(81,94)(82,95)(83,96)(84,97)(85,103)(86,104)(87,105)(88,99)(89,100)(90,101)(91,102)(106,126)(107,120)(108,121)(109,122)(110,123)(111,124)(112,125)(113,131)(114,132)(115,133)(116,127)(117,128)(118,129)(119,130)(134,154)(135,148)(136,149)(137,150)(138,151)(139,152)(140,153)(141,159)(142,160)(143,161)(144,155)(145,156)(146,157)(147,158)(162,182)(163,176)(164,177)(165,178)(166,179)(167,180)(168,181)(169,187)(170,188)(171,189)(172,183)(173,184)(174,185)(175,186)(190,210)(191,204)(192,205)(193,206)(194,207)(195,208)(196,209)(197,215)(198,216)(199,217)(200,211)(201,212)(202,213)(203,214), (1,134,34,158)(2,135,35,159)(3,136,29,160)(4,137,30,161)(5,138,31,155)(6,139,32,156)(7,140,33,157)(8,131,219,107)(9,132,220,108)(10,133,221,109)(11,127,222,110)(12,128,223,111)(13,129,224,112)(14,130,218,106)(15,117,22,124)(16,118,23,125)(17,119,24,126)(18,113,25,120)(19,114,26,121)(20,115,27,122)(21,116,28,123)(36,141,46,148)(37,142,47,149)(38,143,48,150)(39,144,49,151)(40,145,43,152)(41,146,44,153)(42,147,45,154)(50,186,74,162)(51,187,75,163)(52,188,76,164)(53,189,77,165)(54,183,71,166)(55,184,72,167)(56,185,73,168)(57,176,64,169)(58,177,65,170)(59,178,66,171)(60,179,67,172)(61,180,68,173)(62,181,69,174)(63,182,70,175)(78,214,102,190)(79,215,103,191)(80,216,104,192)(81,217,105,193)(82,211,99,194)(83,212,100,195)(84,213,101,196)(85,204,92,197)(86,205,93,198)(87,206,94,199)(88,207,95,200)(89,208,96,201)(90,209,97,202)(91,210,98,203), (1,78,34,102)(2,79,35,103)(3,80,29,104)(4,81,30,105)(5,82,31,99)(6,83,32,100)(7,84,33,101)(8,163,219,187)(9,164,220,188)(10,165,221,189)(11,166,222,183)(12,167,223,184)(13,168,224,185)(14,162,218,186)(15,180,22,173)(16,181,23,174)(17,182,24,175)(18,176,25,169)(19,177,26,170)(20,178,27,171)(21,179,28,172)(36,85,46,92)(37,86,47,93)(38,87,48,94)(39,88,49,95)(40,89,43,96)(41,90,44,97)(42,91,45,98)(50,130,74,106)(51,131,75,107)(52,132,76,108)(53,133,77,109)(54,127,71,110)(55,128,72,111)(56,129,73,112)(57,120,64,113)(58,121,65,114)(59,122,66,115)(60,123,67,116)(61,124,68,117)(62,125,69,118)(63,126,70,119)(134,190,158,214)(135,191,159,215)(136,192,160,216)(137,193,161,217)(138,194,155,211)(139,195,156,212)(140,196,157,213)(141,197,148,204)(142,198,149,205)(143,199,150,206)(144,200,151,207)(145,201,152,208)(146,202,153,209)(147,203,154,210), (1,50,34,74)(2,51,35,75)(3,52,29,76)(4,53,30,77)(5,54,31,71)(6,55,32,72)(7,56,33,73)(8,191,219,215)(9,192,220,216)(10,193,221,217)(11,194,222,211)(12,195,223,212)(13,196,224,213)(14,190,218,214)(15,208,22,201)(16,209,23,202)(17,210,24,203)(18,204,25,197)(19,205,26,198)(20,206,27,199)(21,207,28,200)(36,57,46,64)(37,58,47,65)(38,59,48,66)(39,60,49,67)(40,61,43,68)(41,62,44,69)(42,63,45,70)(78,126,102,119)(79,120,103,113)(80,121,104,114)(81,122,105,115)(82,123,99,116)(83,124,100,117)(84,125,101,118)(85,131,92,107)(86,132,93,108)(87,133,94,109)(88,127,95,110)(89,128,96,111)(90,129,97,112)(91,130,98,106)(134,175,158,182)(135,169,159,176)(136,170,160,177)(137,171,161,178)(138,172,155,179)(139,173,156,180)(140,174,157,181)(141,163,148,187)(142,164,149,188)(143,165,150,189)(144,166,151,183)(145,167,152,184)(146,168,153,185)(147,162,154,186), (8,18)(9,19)(10,20)(11,21)(12,15)(13,16)(14,17)(22,223)(23,224)(24,218)(25,219)(26,220)(27,221)(28,222)(78,102)(79,103)(80,104)(81,105)(82,99)(83,100)(84,101)(85,92)(86,93)(87,94)(88,95)(89,96)(90,97)(91,98)(106,130)(107,131)(108,132)(109,133)(110,127)(111,128)(112,129)(113,120)(114,121)(115,122)(116,123)(117,124)(118,125)(119,126)(134,147)(135,141)(136,142)(137,143)(138,144)(139,145)(140,146)(148,159)(149,160)(150,161)(151,155)(152,156)(153,157)(154,158)(162,175)(163,169)(164,170)(165,171)(166,172)(167,173)(168,174)(176,187)(177,188)(178,189)(179,183)(180,184)(181,185)(182,186)(190,210)(191,204)(192,205)(193,206)(194,207)(195,208)(196,209)(197,215)(198,216)(199,217)(200,211)(201,212)(202,213)(203,214)>;
 
G:=Group( (1,2,3,4,5,6,7)(8,9,10,11,12,13,14)(15,16,17,18,19,20,21)(22,23,24,25,26,27,28)(29,30,31,32,33,34,35)(36,37,38,39,40,41,42)(43,44,45,46,47,48,49)(50,51,52,53,54,55,56)(57,58,59,60,61,62,63)(64,65,66,67,68,69,70)(71,72,73,74,75,76,77)(78,79,80,81,82,83,84)(85,86,87,88,89,90,91)(92,93,94,95,96,97,98)(99,100,101,102,103,104,105)(106,107,108,109,110,111,112)(113,114,115,116,117,118,119)(120,121,122,123,124,125,126)(127,128,129,130,131,132,133)(134,135,136,137,138,139,140)(141,142,143,144,145,146,147)(148,149,150,151,152,153,154)(155,156,157,158,159,160,161)(162,163,164,165,166,167,168)(169,170,171,172,173,174,175)(176,177,178,179,180,181,182)(183,184,185,186,187,188,189)(190,191,192,193,194,195,196)(197,198,199,200,201,202,203)(204,205,206,207,208,209,210)(211,212,213,214,215,216,217)(218,219,220,221,222,223,224), (1,34)(2,35)(3,29)(4,30)(5,31)(6,32)(7,33)(8,219)(9,220)(10,221)(11,222)(12,223)(13,224)(14,218)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28)(36,46)(37,47)(38,48)(39,49)(40,43)(41,44)(42,45)(50,74)(51,75)(52,76)(53,77)(54,71)(55,72)(56,73)(57,64)(58,65)(59,66)(60,67)(61,68)(62,69)(63,70)(78,102)(79,103)(80,104)(81,105)(82,99)(83,100)(84,101)(85,92)(86,93)(87,94)(88,95)(89,96)(90,97)(91,98)(106,130)(107,131)(108,132)(109,133)(110,127)(111,128)(112,129)(113,120)(114,121)(115,122)(116,123)(117,124)(118,125)(119,126)(134,158)(135,159)(136,160)(137,161)(138,155)(139,156)(140,157)(141,148)(142,149)(143,150)(144,151)(145,152)(146,153)(147,154)(162,186)(163,187)(164,188)(165,189)(166,183)(167,184)(168,185)(169,176)(170,177)(171,178)(172,179)(173,180)(174,181)(175,182)(190,214)(191,215)(192,216)(193,217)(194,211)(195,212)(196,213)(197,204)(198,205)(199,206)(200,207)(201,208)(202,209)(203,210), (1,45)(2,46)(3,47)(4,48)(5,49)(6,43)(7,44)(8,18)(9,19)(10,20)(11,21)(12,15)(13,16)(14,17)(22,223)(23,224)(24,218)(25,219)(26,220)(27,221)(28,222)(29,37)(30,38)(31,39)(32,40)(33,41)(34,42)(35,36)(50,70)(51,64)(52,65)(53,66)(54,67)(55,68)(56,69)(57,75)(58,76)(59,77)(60,71)(61,72)(62,73)(63,74)(78,98)(79,92)(80,93)(81,94)(82,95)(83,96)(84,97)(85,103)(86,104)(87,105)(88,99)(89,100)(90,101)(91,102)(106,126)(107,120)(108,121)(109,122)(110,123)(111,124)(112,125)(113,131)(114,132)(115,133)(116,127)(117,128)(118,129)(119,130)(134,154)(135,148)(136,149)(137,150)(138,151)(139,152)(140,153)(141,159)(142,160)(143,161)(144,155)(145,156)(146,157)(147,158)(162,182)(163,176)(164,177)(165,178)(166,179)(167,180)(168,181)(169,187)(170,188)(171,189)(172,183)(173,184)(174,185)(175,186)(190,210)(191,204)(192,205)(193,206)(194,207)(195,208)(196,209)(197,215)(198,216)(199,217)(200,211)(201,212)(202,213)(203,214), (1,134,34,158)(2,135,35,159)(3,136,29,160)(4,137,30,161)(5,138,31,155)(6,139,32,156)(7,140,33,157)(8,131,219,107)(9,132,220,108)(10,133,221,109)(11,127,222,110)(12,128,223,111)(13,129,224,112)(14,130,218,106)(15,117,22,124)(16,118,23,125)(17,119,24,126)(18,113,25,120)(19,114,26,121)(20,115,27,122)(21,116,28,123)(36,141,46,148)(37,142,47,149)(38,143,48,150)(39,144,49,151)(40,145,43,152)(41,146,44,153)(42,147,45,154)(50,186,74,162)(51,187,75,163)(52,188,76,164)(53,189,77,165)(54,183,71,166)(55,184,72,167)(56,185,73,168)(57,176,64,169)(58,177,65,170)(59,178,66,171)(60,179,67,172)(61,180,68,173)(62,181,69,174)(63,182,70,175)(78,214,102,190)(79,215,103,191)(80,216,104,192)(81,217,105,193)(82,211,99,194)(83,212,100,195)(84,213,101,196)(85,204,92,197)(86,205,93,198)(87,206,94,199)(88,207,95,200)(89,208,96,201)(90,209,97,202)(91,210,98,203), (1,78,34,102)(2,79,35,103)(3,80,29,104)(4,81,30,105)(5,82,31,99)(6,83,32,100)(7,84,33,101)(8,163,219,187)(9,164,220,188)(10,165,221,189)(11,166,222,183)(12,167,223,184)(13,168,224,185)(14,162,218,186)(15,180,22,173)(16,181,23,174)(17,182,24,175)(18,176,25,169)(19,177,26,170)(20,178,27,171)(21,179,28,172)(36,85,46,92)(37,86,47,93)(38,87,48,94)(39,88,49,95)(40,89,43,96)(41,90,44,97)(42,91,45,98)(50,130,74,106)(51,131,75,107)(52,132,76,108)(53,133,77,109)(54,127,71,110)(55,128,72,111)(56,129,73,112)(57,120,64,113)(58,121,65,114)(59,122,66,115)(60,123,67,116)(61,124,68,117)(62,125,69,118)(63,126,70,119)(134,190,158,214)(135,191,159,215)(136,192,160,216)(137,193,161,217)(138,194,155,211)(139,195,156,212)(140,196,157,213)(141,197,148,204)(142,198,149,205)(143,199,150,206)(144,200,151,207)(145,201,152,208)(146,202,153,209)(147,203,154,210), (1,50,34,74)(2,51,35,75)(3,52,29,76)(4,53,30,77)(5,54,31,71)(6,55,32,72)(7,56,33,73)(8,191,219,215)(9,192,220,216)(10,193,221,217)(11,194,222,211)(12,195,223,212)(13,196,224,213)(14,190,218,214)(15,208,22,201)(16,209,23,202)(17,210,24,203)(18,204,25,197)(19,205,26,198)(20,206,27,199)(21,207,28,200)(36,57,46,64)(37,58,47,65)(38,59,48,66)(39,60,49,67)(40,61,43,68)(41,62,44,69)(42,63,45,70)(78,126,102,119)(79,120,103,113)(80,121,104,114)(81,122,105,115)(82,123,99,116)(83,124,100,117)(84,125,101,118)(85,131,92,107)(86,132,93,108)(87,133,94,109)(88,127,95,110)(89,128,96,111)(90,129,97,112)(91,130,98,106)(134,175,158,182)(135,169,159,176)(136,170,160,177)(137,171,161,178)(138,172,155,179)(139,173,156,180)(140,174,157,181)(141,163,148,187)(142,164,149,188)(143,165,150,189)(144,166,151,183)(145,167,152,184)(146,168,153,185)(147,162,154,186), (8,18)(9,19)(10,20)(11,21)(12,15)(13,16)(14,17)(22,223)(23,224)(24,218)(25,219)(26,220)(27,221)(28,222)(78,102)(79,103)(80,104)(81,105)(82,99)(83,100)(84,101)(85,92)(86,93)(87,94)(88,95)(89,96)(90,97)(91,98)(106,130)(107,131)(108,132)(109,133)(110,127)(111,128)(112,129)(113,120)(114,121)(115,122)(116,123)(117,124)(118,125)(119,126)(134,147)(135,141)(136,142)(137,143)(138,144)(139,145)(140,146)(148,159)(149,160)(150,161)(151,155)(152,156)(153,157)(154,158)(162,175)(163,169)(164,170)(165,171)(166,172)(167,173)(168,174)(176,187)(177,188)(178,189)(179,183)(180,184)(181,185)(182,186)(190,210)(191,204)(192,205)(193,206)(194,207)(195,208)(196,209)(197,215)(198,216)(199,217)(200,211)(201,212)(202,213)(203,214) );
 
G=PermutationGroup([[(1,2,3,4,5,6,7),(8,9,10,11,12,13,14),(15,16,17,18,19,20,21),(22,23,24,25,26,27,28),(29,30,31,32,33,34,35),(36,37,38,39,40,41,42),(43,44,45,46,47,48,49),(50,51,52,53,54,55,56),(57,58,59,60,61,62,63),(64,65,66,67,68,69,70),(71,72,73,74,75,76,77),(78,79,80,81,82,83,84),(85,86,87,88,89,90,91),(92,93,94,95,96,97,98),(99,100,101,102,103,104,105),(106,107,108,109,110,111,112),(113,114,115,116,117,118,119),(120,121,122,123,124,125,126),(127,128,129,130,131,132,133),(134,135,136,137,138,139,140),(141,142,143,144,145,146,147),(148,149,150,151,152,153,154),(155,156,157,158,159,160,161),(162,163,164,165,166,167,168),(169,170,171,172,173,174,175),(176,177,178,179,180,181,182),(183,184,185,186,187,188,189),(190,191,192,193,194,195,196),(197,198,199,200,201,202,203),(204,205,206,207,208,209,210),(211,212,213,214,215,216,217),(218,219,220,221,222,223,224)], [(1,34),(2,35),(3,29),(4,30),(5,31),(6,32),(7,33),(8,219),(9,220),(10,221),(11,222),(12,223),(13,224),(14,218),(15,22),(16,23),(17,24),(18,25),(19,26),(20,27),(21,28),(36,46),(37,47),(38,48),(39,49),(40,43),(41,44),(42,45),(50,74),(51,75),(52,76),(53,77),(54,71),(55,72),(56,73),(57,64),(58,65),(59,66),(60,67),(61,68),(62,69),(63,70),(78,102),(79,103),(80,104),(81,105),(82,99),(83,100),(84,101),(85,92),(86,93),(87,94),(88,95),(89,96),(90,97),(91,98),(106,130),(107,131),(108,132),(109,133),(110,127),(111,128),(112,129),(113,120),(114,121),(115,122),(116,123),(117,124),(118,125),(119,126),(134,158),(135,159),(136,160),(137,161),(138,155),(139,156),(140,157),(141,148),(142,149),(143,150),(144,151),(145,152),(146,153),(147,154),(162,186),(163,187),(164,188),(165,189),(166,183),(167,184),(168,185),(169,176),(170,177),(171,178),(172,179),(173,180),(174,181),(175,182),(190,214),(191,215),(192,216),(193,217),(194,211),(195,212),(196,213),(197,204),(198,205),(199,206),(200,207),(201,208),(202,209),(203,210)], [(1,45),(2,46),(3,47),(4,48),(5,49),(6,43),(7,44),(8,18),(9,19),(10,20),(11,21),(12,15),(13,16),(14,17),(22,223),(23,224),(24,218),(25,219),(26,220),(27,221),(28,222),(29,37),(30,38),(31,39),(32,40),(33,41),(34,42),(35,36),(50,70),(51,64),(52,65),(53,66),(54,67),(55,68),(56,69),(57,75),(58,76),(59,77),(60,71),(61,72),(62,73),(63,74),(78,98),(79,92),(80,93),(81,94),(82,95),(83,96),(84,97),(85,103),(86,104),(87,105),(88,99),(89,100),(90,101),(91,102),(106,126),(107,120),(108,121),(109,122),(110,123),(111,124),(112,125),(113,131),(114,132),(115,133),(116,127),(117,128),(118,129),(119,130),(134,154),(135,148),(136,149),(137,150),(138,151),(139,152),(140,153),(141,159),(142,160),(143,161),(144,155),(145,156),(146,157),(147,158),(162,182),(163,176),(164,177),(165,178),(166,179),(167,180),(168,181),(169,187),(170,188),(171,189),(172,183),(173,184),(174,185),(175,186),(190,210),(191,204),(192,205),(193,206),(194,207),(195,208),(196,209),(197,215),(198,216),(199,217),(200,211),(201,212),(202,213),(203,214)], [(1,134,34,158),(2,135,35,159),(3,136,29,160),(4,137,30,161),(5,138,31,155),(6,139,32,156),(7,140,33,157),(8,131,219,107),(9,132,220,108),(10,133,221,109),(11,127,222,110),(12,128,223,111),(13,129,224,112),(14,130,218,106),(15,117,22,124),(16,118,23,125),(17,119,24,126),(18,113,25,120),(19,114,26,121),(20,115,27,122),(21,116,28,123),(36,141,46,148),(37,142,47,149),(38,143,48,150),(39,144,49,151),(40,145,43,152),(41,146,44,153),(42,147,45,154),(50,186,74,162),(51,187,75,163),(52,188,76,164),(53,189,77,165),(54,183,71,166),(55,184,72,167),(56,185,73,168),(57,176,64,169),(58,177,65,170),(59,178,66,171),(60,179,67,172),(61,180,68,173),(62,181,69,174),(63,182,70,175),(78,214,102,190),(79,215,103,191),(80,216,104,192),(81,217,105,193),(82,211,99,194),(83,212,100,195),(84,213,101,196),(85,204,92,197),(86,205,93,198),(87,206,94,199),(88,207,95,200),(89,208,96,201),(90,209,97,202),(91,210,98,203)], [(1,78,34,102),(2,79,35,103),(3,80,29,104),(4,81,30,105),(5,82,31,99),(6,83,32,100),(7,84,33,101),(8,163,219,187),(9,164,220,188),(10,165,221,189),(11,166,222,183),(12,167,223,184),(13,168,224,185),(14,162,218,186),(15,180,22,173),(16,181,23,174),(17,182,24,175),(18,176,25,169),(19,177,26,170),(20,178,27,171),(21,179,28,172),(36,85,46,92),(37,86,47,93),(38,87,48,94),(39,88,49,95),(40,89,43,96),(41,90,44,97),(42,91,45,98),(50,130,74,106),(51,131,75,107),(52,132,76,108),(53,133,77,109),(54,127,71,110),(55,128,72,111),(56,129,73,112),(57,120,64,113),(58,121,65,114),(59,122,66,115),(60,123,67,116),(61,124,68,117),(62,125,69,118),(63,126,70,119),(134,190,158,214),(135,191,159,215),(136,192,160,216),(137,193,161,217),(138,194,155,211),(139,195,156,212),(140,196,157,213),(141,197,148,204),(142,198,149,205),(143,199,150,206),(144,200,151,207),(145,201,152,208),(146,202,153,209),(147,203,154,210)], [(1,50,34,74),(2,51,35,75),(3,52,29,76),(4,53,30,77),(5,54,31,71),(6,55,32,72),(7,56,33,73),(8,191,219,215),(9,192,220,216),(10,193,221,217),(11,194,222,211),(12,195,223,212),(13,196,224,213),(14,190,218,214),(15,208,22,201),(16,209,23,202),(17,210,24,203),(18,204,25,197),(19,205,26,198),(20,206,27,199),(21,207,28,200),(36,57,46,64),(37,58,47,65),(38,59,48,66),(39,60,49,67),(40,61,43,68),(41,62,44,69),(42,63,45,70),(78,126,102,119),(79,120,103,113),(80,121,104,114),(81,122,105,115),(82,123,99,116),(83,124,100,117),(84,125,101,118),(85,131,92,107),(86,132,93,108),(87,133,94,109),(88,127,95,110),(89,128,96,111),(90,129,97,112),(91,130,98,106),(134,175,158,182),(135,169,159,176),(136,170,160,177),(137,171,161,178),(138,172,155,179),(139,173,156,180),(140,174,157,181),(141,163,148,187),(142,164,149,188),(143,165,150,189),(144,166,151,183),(145,167,152,184),(146,168,153,185),(147,162,154,186)], [(8,18),(9,19),(10,20),(11,21),(12,15),(13,16),(14,17),(22,223),(23,224),(24,218),(25,219),(26,220),(27,221),(28,222),(78,102),(79,103),(80,104),(81,105),(82,99),(83,100),(84,101),(85,92),(86,93),(87,94),(88,95),(89,96),(90,97),(91,98),(106,130),(107,131),(108,132),(109,133),(110,127),(111,128),(112,129),(113,120),(114,121),(115,122),(116,123),(117,124),(118,125),(119,126),(134,147),(135,141),(136,142),(137,143),(138,144),(139,145),(140,146),(148,159),(149,160),(150,161),(151,155),(152,156),(153,157),(154,158),(162,175),(163,169),(164,170),(165,171),(166,172),(167,173),(168,174),(176,187),(177,188),(178,189),(179,183),(180,184),(181,185),(182,186),(190,210),(191,204),(192,205),(193,206),(194,207),(195,208),(196,209),(197,215),(198,216),(199,217),(200,211),(201,212),(202,213),(203,214)]])
 

133 conjugacy classes

class 1 2A2B2C2D2E4A···4M7A···7F14A···14R14S···14AD28A···28BZ
order1222224···47···714···1414···1428···28
size1111444···41···11···14···44···4

133 irreducible representations

dim111111111111114444
type++++++++-
imageC1C2C2C2C2C2C2C7C14C14C14C14C14C142+ 1+42- 1+4C7×2+ 1+4C7×2- 1+4
kernelC7×C22.57C24C7×C22⋊Q8C7×C22.D4C7×C4.4D4C7×C42.C2C7×C42⋊2C2C7×C4⋊Q8C22.57C24C22⋊Q8C22.D4C4.4D4C42.C2C42⋊2C2C4⋊Q8C14C14C2C2
# reps142124262412612241212612

Matrix representation of C7×C22.57C24 ►in GL8(𝔽29)

200000000
020000000
002000000
000200000
000016000
000001600
000000160
000000016
,
10000000
01000000
00100000
00010000
000028000
000002800
000000280
000000028
,
280000000
028000000
002800000
000280000
00001000
00000100
00000010
00000001
,
71911100000
20209270000
27022100000
027990000
000099911
00001111277
000018900
0000110189
,
11210190000
271825190000
001120000
0027180000
00000010
0000112827
000028000
000011028
,
015280000
102850000
000280000
002800000
00000100
000028000
0000282812
0000012828
,
107190000
0120200000
002800000
000280000
00001000
00000100
000000280
000011028

G:=sub<GL(8,GF(29))| [20,0,0,0,0,0,0,0,0,20,0,0,0,0,0,0,0,0,20,0,0,0,0,0,0,0,0,20,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28],[28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[7,20,27,0,0,0,0,0,19,20,0,27,0,0,0,0,11,9,22,9,0,0,0,0,10,27,10,9,0,0,0,0,0,0,0,0,9,11,18,11,0,0,0,0,9,11,9,0,0,0,0,0,9,27,0,18,0,0,0,0,11,7,0,9],[11,27,0,0,0,0,0,0,2,18,0,0,0,0,0,0,10,25,11,27,0,0,0,0,19,19,2,18,0,0,0,0,0,0,0,0,0,1,28,1,0,0,0,0,0,1,0,1,0,0,0,0,1,28,0,0,0,0,0,0,0,27,0,28],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,5,28,0,28,0,0,0,0,28,5,28,0,0,0,0,0,0,0,0,0,0,28,28,0,0,0,0,0,1,0,28,1,0,0,0,0,0,0,1,28,0,0,0,0,0,0,2,28],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,7,20,28,0,0,0,0,0,19,20,0,28,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28] >;
 

C7×C22.57C24 in GAP, Magma, Sage, TeX 

C_7\times C_2^2._{57}C_2^4
 
% in TeX
 
G:=Group("C7xC2^2.57C2^4");
 
// GroupNames label
 
G:=SmallGroup(448,1346);
 
// by ID
 
G=gap.SmallGroup(448,1346);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-7,-2,-2,784,1597,792,4790,3579,604,9635,1690]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^7=b^2=c^2=g^2=1,d^2=e^2=f^2=b,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,b*c=c*b,e*d*e^-1=b*d=d*b,g*e*g=b*e=e*b,b*f=f*b,b*g=g*b,f*d*f^-1=c*d=d*c,c*e=e*c,c*f=f*c,c*g=g*c,g*d*g=b*c*d,f*e*f^-1=b*c*e,f*g=g*f>;
 
// generators/relations
 

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