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G = C7×C23.32C23  order 448 = 26·7

Direct product of C7 and C23.32C23

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

Aliases: C7×C23.32C23, C14.1092- 1+4, (C2×Q8)⋊9C28, (C4×Q8)⋊5C14, (Q8×C28)⋊25C2, (Q8×C14)⋊19C4, Q8.8(C2×C28), C2.8(C23×C28), C4.20(C22×C28), C42.32(C2×C14), C14.60(C23×C4), (C22×Q8).7C14, (C4×C28).275C22, (C2×C14).339C24, C28.165(C22×C4), (C2×C28).710C23, C42⋊C2.10C14, C2.1(C7×2- 1+4), C22.11(C22×C28), C22.12(C23×C14), C23.31(C22×C14), (Q8×C14).284C22, (C22×C28).442C22, (C22×C14).255C23, (Q8×C2×C14).17C2, C4⋊C4.82(C2×C14), (C2×C4).31(C2×C28), (C7×Q8).32(C2×C4), (C2×C28).203(C2×C4), (C2×Q8).72(C2×C14), (C7×C4⋊C4).407C22, C22⋊C4.29(C2×C14), (C22×C4).52(C2×C14), (C2×C14).165(C22×C4), (C2×C4).135(C22×C14), (C7×C42⋊C2).24C2, (C7×C22⋊C4).160C22, SmallGroup(448,1302)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2 — C7×C23.32C23
Chief series
C1 — C2 — C22 — C2×C14 — C2×C28 — C7×C4⋊C4 — Q8×C28 — C7×C23.32C23
Lower central
C1 — C2 — C7×C23.32C23
Upper central
C1 — C2×C14 — C7×C23.32C23

Generators and relations for C7×C23.32C23
 G = < a,b,c,d,e,f,g | a7=b2=c2=d2=1, e2=d, f2=g2=c, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, ebe-1=bc=cb, bd=db, bf=fb, bg=gb, cd=dc, fef-1=ce=ec, gfg-1=cf=fc, cg=gc, de=ed, df=fd, dg=gd, eg=ge >

Subgroups: 290 in 266 conjugacy classes, 242 normal (12 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C7, C2×C4, Q8, C23, C14, C14, C14, C42, C22⋊C4, C4⋊C4, C22×C4, C2×Q8, C28, C28, C2×C14, C2×C14, C2×C14, C42⋊C2, C4×Q8, C22×Q8, C2×C28, C7×Q8, C22×C14, C23.32C23, C4×C28, C7×C22⋊C4, C7×C4⋊C4, C22×C28, Q8×C14, C7×C42⋊C2, Q8×C28, Q8×C2×C14, C7×C23.32C23
Quotients: C1, C2, C4, C22, C7, C2×C4, C23, C14, C22×C4, C24, C28, C2×C14, C23×C4, 2- 1+4, C2×C28, C22×C14, C23.32C23, C22×C28, C23×C14, C23×C28, C7×2- 1+4, C7×C23.32C23

Smallest permutation representation of C7×C23.32C23
►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 35)(2 29)(3 30)(4 31)(5 32)(6 33)(7 34)(8 224)(9 218)(10 219)(11 220)(12 221)(13 222)(14 223)(15 27)(16 28)(17 22)(18 23)(19 24)(20 25)(21 26)(36 44)(37 45)(38 46)(39 47)(40 48)(41 49)(42 43)(50 75)(51 76)(52 77)(53 71)(54 72)(55 73)(56 74)(57 67)(58 68)(59 69)(60 70)(61 64)(62 65)(63 66)(78 103)(79 104)(80 105)(81 99)(82 100)(83 101)(84 102)(85 95)(86 96)(87 97)(88 98)(89 92)(90 93)(91 94)(106 131)(107 132)(108 133)(109 127)(110 128)(111 129)(112 130)(113 123)(114 124)(115 125)(116 126)(117 120)(118 121)(119 122)(134 147)(135 141)(136 142)(137 143)(138 144)(139 145)(140 146)(148 157)(149 158)(150 159)(151 160)(152 161)(153 155)(154 156)(162 175)(163 169)(164 170)(165 171)(166 172)(167 173)(168 174)(176 185)(177 186)(178 187)(179 188)(180 189)(181 183)(182 184)(190 203)(191 197)(192 198)(193 199)(194 200)(195 201)(196 202)(204 213)(205 214)(206 215)(207 216)(208 217)(209 211)(210 212)

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

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

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

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

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

238 conjugacy classes

class 1 2A2B2C2D2E4A···4AB7A···7F14A···14R14S···14AD28A···28FL
order1222224···47···714···1414···1428···28
size1111222···21···11···12···22···2

238 irreducible representations

dim111111111144
type++++-
imageC1C2C2C2C4C7C14C14C14C282- 1+4C7×2- 1+4
kernelC7×C23.32C23C7×C42⋊C2Q8×C28Q8×C2×C14Q8×C14C23.32C23C42⋊C2C4×Q8C22×Q8C2×Q8C14C2
# reps16811663648696212

Matrix representation of C7×C23.32C23 ►in GL5(𝔽29)

200000
07000
00700
00070
00007
,
10000
010010
00109
000280
000028
,
10000
028000
002800
000280
000028
,
280000
01000
00100
00010
00001
,
170000
0019118
0020018
0119018
00209
,
280000
028271012
01191
000113
0002718
,
10000
017504
001200
0001713
000012

G:=sub<GL(5,GF(29))| [20,0,0,0,0,0,7,0,0,0,0,0,7,0,0,0,0,0,7,0,0,0,0,0,7],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,28,0,0,10,9,0,28],[1,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,28],[28,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[17,0,0,0,0,0,0,0,1,0,0,19,20,19,2,0,1,0,0,0,0,18,18,18,9],[28,0,0,0,0,0,28,1,0,0,0,27,1,0,0,0,10,9,11,27,0,12,1,3,18],[1,0,0,0,0,0,17,0,0,0,0,5,12,0,0,0,0,0,17,0,0,4,0,13,12] >;
 

C7×C23.32C23 in GAP, Magma, Sage, TeX 

C_7\times C_2^3._{32}C_2^3
 
% in TeX
 
G:=Group("C7xC2^3.32C2^3");
 
// GroupNames label
 
G:=SmallGroup(448,1302);
 
// by ID
 
G=gap.SmallGroup(448,1302);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-7,-2,-2,1568,1597,792,1227,604,3363]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^7=b^2=c^2=d^2=1,e^2=d,f^2=g^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,e*b*e^-1=b*c=c*b,b*d=d*b,b*f=f*b,b*g=g*b,c*d=d*c,f*e*f^-1=c*e=e*c,g*f*g^-1=c*f=f*c,c*g=g*c,d*e=e*d,d*f=f*d,d*g=g*d,e*g=g*e>;
 
// generators/relations
 

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