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G = C7×C22.35C24  order 448 = 26·7

Direct product of C7 and C22.35C24

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

Aliases: C7×C22.35C24, C14.1142- 1+4, C4⋊Q8⋊10C14, (C4×Q8)⋊9C14, (Q8×C28)⋊29C2, C42⋊2C2.C14, C42.C2⋊6C14, C22⋊Q8.8C14, C42.39(C2×C14), C28.277(C4○D4), (C2×C14).361C24, (C2×C28).670C23, (C4×C28).280C22, C42⋊C2.12C14, C2.6(C7×2- 1+4), C23.13(C22×C14), C22.35(C23×C14), (C22×C14).96C23, (Q8×C14).272C22, (C22×C28).449C22, (C7×C4⋊Q8)⋊31C2, C4.21(C7×C4○D4), C4⋊C4.69(C2×C14), C2.18(C14×C4○D4), C14.237(C2×C4○D4), C22⋊C4.3(C2×C14), (C2×Q8).59(C2×C14), (C7×C42.C2)⋊23C2, (C7×C22⋊Q8).18C2, (C7×C4⋊C4).248C22, (C2×C4).28(C22×C14), (C22×C4).61(C2×C14), (C7×C42⋊2C2).2C2, (C7×C42⋊C2).26C2, (C7×C22⋊C4).149C22, SmallGroup(448,1324)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C22 — C7×C22.35C24
Chief series
C1 — C2 — C22 — C2×C14 — C2×C28 — C7×C4⋊C4 — C7×C42⋊2C2 — C7×C22.35C24
Lower central
C1 — C22 — C7×C22.35C24
Upper central
C1 — C2×C14 — C7×C22.35C24

Generators and relations for C7×C22.35C24
 G = < a,b,c,d,e,f,g | a7=b2=c2=f2=1, d2=g2=b, e2=c, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, ede-1=gdg-1=bd=db, fef=be=eb, bf=fb, bg=gb, fdf=cd=dc, ce=ec, cf=fc, cg=gc, eg=ge, fg=gf >

Subgroups: 242 in 192 conjugacy classes, 146 normal (26 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C7, C2×C4, C2×C4, C2×C4, Q8, C23, C14, C14, C14, C42, C42, C22⋊C4, C4⋊C4, C22×C4, C2×Q8, C28, C28, C2×C14, C2×C14, C42⋊C2, C4×Q8, C22⋊Q8, C42.C2, C42.C2, C42⋊2C2, C4⋊Q8, C2×C28, C2×C28, C2×C28, C7×Q8, C22×C14, C22.35C24, C4×C28, C4×C28, C7×C22⋊C4, C7×C4⋊C4, C22×C28, Q8×C14, C7×C42⋊C2, Q8×C28, C7×C22⋊Q8, C7×C42.C2, C7×C42.C2, C7×C42⋊2C2, C7×C4⋊Q8, C7×C22.35C24
Quotients: C1, C2, C22, C7, C23, C14, C4○D4, C24, C2×C14, C2×C4○D4, 2- 1+4, C22×C14, C22.35C24, C7×C4○D4, C23×C14, C14×C4○D4, C7×2- 1+4, C7×C22.35C24

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

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

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

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

(8 224)(9 218)(10 219)(11 220)(12 221)(13 222)(14 223)(15 24)(16 25)(17 26)(18 27)(19 28)(20 22)(21 23)(78 94)(79 95)(80 96)(81 97)(82 98)(83 92)(84 93)(85 100)(86 101)(87 102)(88 103)(89 104)(90 105)(91 99)(106 122)(107 123)(108 124)(109 125)(110 126)(111 120)(112 121)(113 128)(114 129)(115 130)(116 131)(117 132)(118 133)(119 127)(134 155)(135 156)(136 157)(137 158)(138 159)(139 160)(140 161)(141 151)(142 152)(143 153)(144 154)(145 148)(146 149)(147 150)(162 183)(163 184)(164 185)(165 186)(166 187)(167 188)(168 189)(169 179)(170 180)(171 181)(172 182)(173 176)(174 177)(175 178)(190 203)(191 197)(192 198)(193 199)(194 200)(195 201)(196 202)(204 216)(205 217)(206 211)(207 212)(208 213)(209 214)(210 215)

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

154 conjugacy classes

class 1 2A2B2C2D4A···4F4G···4Q7A···7F14A···14R14S···14X28A···28AJ28AK···28CX
order122224···44···47···714···1414···1428···2828···28
size111142···24···41···11···14···42···24···4

154 irreducible representations

dim111111111111112244
type+++++++-
imageC1C2C2C2C2C2C2C7C14C14C14C14C14C14C4○D4C7×C4○D42- 1+4C7×2- 1+4
kernelC7×C22.35C24C7×C42⋊C2Q8×C28C7×C22⋊Q8C7×C42.C2C7×C42⋊2C2C7×C4⋊Q8C22.35C24C42⋊C2C4×Q8C22⋊Q8C42.C2C42⋊2C2C4⋊Q8C28C4C14C2
# reps112254166121230246424212

Matrix representation of C7×C22.35C24 ►in GL6(𝔽29)

100000
010000
0024000
0002400
0000240
0000024
,
100000
010000
0028000
0002800
0000280
0000028
,
2800000
0280000
001000
000100
000010
000001
,
1210000
2170000
00121600
0001700
001721158
002627814
,
1200000
0120000
00280270
000071
000010
0001220
,
100000
5280000
001000
000100
00280280
0070028
,
100000
010000
00222700
0025700
000101
00422280

G:=sub<GL(6,GF(29))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,24,0,0,0,0,0,0,24,0,0,0,0,0,0,24,0,0,0,0,0,0,24],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,2,0,0,0,0,1,17,0,0,0,0,0,0,12,0,17,26,0,0,16,17,21,27,0,0,0,0,15,8,0,0,0,0,8,14],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,1,0,0,27,7,1,22,0,0,0,1,0,0],[1,5,0,0,0,0,0,28,0,0,0,0,0,0,1,0,28,7,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,22,25,0,4,0,0,27,7,1,22,0,0,0,0,0,28,0,0,0,0,1,0] >;
 

C7×C22.35C24 in GAP, Magma, Sage, TeX 

C_7\times C_2^2._{35}C_2^4
 
% in TeX
 
G:=Group("C7xC2^2.35C2^4");
 
// GroupNames label
 
G:=SmallGroup(448,1324);
 
// by ID
 
G=gap.SmallGroup(448,1324);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-7,-2,-2,784,1597,1576,4790,1227,3363,416]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^7=b^2=c^2=f^2=1,d^2=g^2=b,e^2=c,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=g*d*g^-1=b*d=d*b,f*e*f=b*e=e*b,b*f=f*b,b*g=g*b,f*d*f=c*d=d*c,c*e=e*c,c*f=f*c,c*g=g*c,e*g=g*e,f*g=g*f>;
 
// generators/relations
 

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