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G = C62.117D4order 288 = 25·32

22nd non-split extension by C62 of D4 acting via D4/C22=C2

metabelian, supersoluble, monomial

Aliases: C62.117D4, (C3×Q8)⋊3Dic3, (C3×C12).58D4, (C2×C12).95D6, (C6×Q8).18S3, (C3×C6).17Q16, (Q8×C32)⋊6C4, Q82(C3⋊Dic3), (C3×C6).30SD16, C33(Q82Dic3), C12.40(C3⋊D4), (C6×C12).62C22, C12.14(C2×Dic3), C6.12(C3⋊Q16), C2.6(C625C4), C6.13(Q82S3), C3213(Q8⋊C4), C4.14(C327D4), C2.3(C327Q16), C12⋊Dic3.14C2, C6.26(C6.D4), C2.3(C3211SD16), C22.18(C327D4), (Q8×C3×C6).3C2, C4.2(C2×C3⋊Dic3), (C3×C12).53(C2×C4), (C2×Q8).3(C3⋊S3), (C2×C6).93(C3⋊D4), (C2×C324C8).9C2, (C3×C6).74(C22⋊C4), (C2×C4).41(C2×C3⋊S3), SmallGroup(288,310)

Series: Derived Chief Lower central Upper central

C1C3×C12 — C62.117D4
C1C3C32C3×C6C62C6×C12C12⋊Dic3 — C62.117D4
C32C3×C6C3×C12 — C62.117D4
C1C22C2×C4C2×Q8

Generators and relations for C62.117D4
 G = < a,b,c,d | a6=b6=1, c4=b3, d2=a3, ab=ba, cac-1=dad-1=a-1, cbc-1=dbd-1=b-1, dcd-1=a3c3 >

Subgroups: 356 in 126 conjugacy classes, 69 normal (21 characteristic)
C1, C2 [×3], C3 [×4], C4 [×2], C4 [×3], C22, C6 [×12], C8, C2×C4, C2×C4 [×2], Q8 [×2], Q8, C32, Dic3 [×4], C12 [×8], C12 [×8], C2×C6 [×4], C4⋊C4, C2×C8, C2×Q8, C3×C6 [×3], C3⋊C8 [×4], C2×Dic3 [×4], C2×C12 [×4], C2×C12 [×4], C3×Q8 [×8], C3×Q8 [×4], Q8⋊C4, C3⋊Dic3, C3×C12 [×2], C3×C12 [×2], C62, C2×C3⋊C8 [×4], C4⋊Dic3 [×4], C6×Q8 [×4], C324C8, C2×C3⋊Dic3, C6×C12, C6×C12, Q8×C32 [×2], Q8×C32, Q82Dic3 [×4], C2×C324C8, C12⋊Dic3, Q8×C3×C6, C62.117D4
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×4], C2×C4, D4 [×2], Dic3 [×8], D6 [×4], C22⋊C4, SD16, Q16, C3⋊S3, C2×Dic3 [×4], C3⋊D4 [×8], Q8⋊C4, C3⋊Dic3 [×2], C2×C3⋊S3, Q82S3 [×4], C3⋊Q16 [×4], C6.D4 [×4], C2×C3⋊Dic3, C327D4 [×2], Q82Dic3 [×4], C3211SD16, C327Q16, C625C4, C62.117D4

Smallest permutation representation of C62.117D4
Regular action on 288 points
Generators in S288
(1 11 285 170 181 102)(2 103 182 171 286 12)(3 13 287 172 183 104)(4 97 184 173 288 14)(5 15 281 174 177 98)(6 99 178 175 282 16)(7 9 283 176 179 100)(8 101 180 169 284 10)(17 243 40 128 92 85)(18 86 93 121 33 244)(19 245 34 122 94 87)(20 88 95 123 35 246)(21 247 36 124 96 81)(22 82 89 125 37 248)(23 241 38 126 90 83)(24 84 91 127 39 242)(25 209 154 151 202 263)(26 264 203 152 155 210)(27 211 156 145 204 257)(28 258 205 146 157 212)(29 213 158 147 206 259)(30 260 207 148 159 214)(31 215 160 149 208 261)(32 262 201 150 153 216)(41 141 165 130 250 233)(42 234 251 131 166 142)(43 143 167 132 252 235)(44 236 253 133 168 144)(45 137 161 134 254 237)(46 238 255 135 162 138)(47 139 163 136 256 239)(48 240 249 129 164 140)(49 228 62 105 79 192)(50 185 80 106 63 229)(51 230 64 107 73 186)(52 187 74 108 57 231)(53 232 58 109 75 188)(54 189 76 110 59 225)(55 226 60 111 77 190)(56 191 78 112 61 227)(65 117 270 275 218 194)(66 195 219 276 271 118)(67 119 272 277 220 196)(68 197 221 278 265 120)(69 113 266 279 222 198)(70 199 223 280 267 114)(71 115 268 273 224 200)(72 193 217 274 269 116)
(1 63 113 5 59 117)(2 118 60 6 114 64)(3 57 115 7 61 119)(4 120 62 8 116 58)(9 227 272 13 231 268)(10 269 232 14 265 228)(11 229 266 15 225 270)(12 271 226 16 267 230)(17 263 130 21 259 134)(18 135 260 22 131 264)(19 257 132 23 261 136)(20 129 262 24 133 258)(25 250 247 29 254 243)(26 244 255 30 248 251)(27 252 241 31 256 245)(28 246 249 32 242 253)(33 238 214 37 234 210)(34 211 235 38 215 239)(35 240 216 39 236 212)(36 213 237 40 209 233)(41 124 158 45 128 154)(42 155 121 46 159 125)(43 126 160 47 122 156)(44 157 123 48 153 127)(49 284 274 53 288 278)(50 279 281 54 275 285)(51 286 276 55 282 280)(52 273 283 56 277 287)(65 102 106 69 98 110)(66 111 99 70 107 103)(67 104 108 71 100 112)(68 105 101 72 109 97)(73 182 195 77 178 199)(74 200 179 78 196 183)(75 184 197 79 180 193)(76 194 181 80 198 177)(81 206 161 85 202 165)(82 166 203 86 162 207)(83 208 163 87 204 167)(84 168 205 88 164 201)(89 142 152 93 138 148)(90 149 139 94 145 143)(91 144 146 95 140 150)(92 151 141 96 147 137)(169 217 188 173 221 192)(170 185 222 174 189 218)(171 219 190 175 223 186)(172 187 224 176 191 220)
(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)(225 226 227 228 229 230 231 232)(233 234 235 236 237 238 239 240)(241 242 243 244 245 246 247 248)(249 250 251 252 253 254 255 256)(257 258 259 260 261 262 263 264)(265 266 267 268 269 270 271 272)(273 274 275 276 277 278 279 280)(281 282 283 284 285 286 287 288)
(1 123 170 20)(2 23 171 126)(3 121 172 18)(4 21 173 124)(5 127 174 24)(6 19 175 122)(7 125 176 22)(8 17 169 128)(9 89 179 248)(10 243 180 92)(11 95 181 246)(12 241 182 90)(13 93 183 244)(14 247 184 96)(15 91 177 242)(16 245 178 94)(25 79 151 228)(26 231 152 74)(27 77 145 226)(28 229 146 80)(29 75 147 232)(30 227 148 78)(31 73 149 230)(32 225 150 76)(33 287 86 104)(34 99 87 282)(35 285 88 102)(36 97 81 288)(37 283 82 100)(38 103 83 286)(39 281 84 98)(40 101 85 284)(41 120 130 221)(42 224 131 115)(43 118 132 219)(44 222 133 113)(45 116 134 217)(46 220 135 119)(47 114 136 223)(48 218 129 117)(49 209 105 202)(50 205 106 212)(51 215 107 208)(52 203 108 210)(53 213 109 206)(54 201 110 216)(55 211 111 204)(56 207 112 214)(57 155 187 264)(58 259 188 158)(59 153 189 262)(60 257 190 156)(61 159 191 260)(62 263 192 154)(63 157 185 258)(64 261 186 160)(65 240 275 164)(66 167 276 235)(67 238 277 162)(68 165 278 233)(69 236 279 168)(70 163 280 239)(71 234 273 166)(72 161 274 237)(137 269 254 193)(138 196 255 272)(139 267 256 199)(140 194 249 270)(141 265 250 197)(142 200 251 268)(143 271 252 195)(144 198 253 266)

G:=sub<Sym(288)| (1,11,285,170,181,102)(2,103,182,171,286,12)(3,13,287,172,183,104)(4,97,184,173,288,14)(5,15,281,174,177,98)(6,99,178,175,282,16)(7,9,283,176,179,100)(8,101,180,169,284,10)(17,243,40,128,92,85)(18,86,93,121,33,244)(19,245,34,122,94,87)(20,88,95,123,35,246)(21,247,36,124,96,81)(22,82,89,125,37,248)(23,241,38,126,90,83)(24,84,91,127,39,242)(25,209,154,151,202,263)(26,264,203,152,155,210)(27,211,156,145,204,257)(28,258,205,146,157,212)(29,213,158,147,206,259)(30,260,207,148,159,214)(31,215,160,149,208,261)(32,262,201,150,153,216)(41,141,165,130,250,233)(42,234,251,131,166,142)(43,143,167,132,252,235)(44,236,253,133,168,144)(45,137,161,134,254,237)(46,238,255,135,162,138)(47,139,163,136,256,239)(48,240,249,129,164,140)(49,228,62,105,79,192)(50,185,80,106,63,229)(51,230,64,107,73,186)(52,187,74,108,57,231)(53,232,58,109,75,188)(54,189,76,110,59,225)(55,226,60,111,77,190)(56,191,78,112,61,227)(65,117,270,275,218,194)(66,195,219,276,271,118)(67,119,272,277,220,196)(68,197,221,278,265,120)(69,113,266,279,222,198)(70,199,223,280,267,114)(71,115,268,273,224,200)(72,193,217,274,269,116), (1,63,113,5,59,117)(2,118,60,6,114,64)(3,57,115,7,61,119)(4,120,62,8,116,58)(9,227,272,13,231,268)(10,269,232,14,265,228)(11,229,266,15,225,270)(12,271,226,16,267,230)(17,263,130,21,259,134)(18,135,260,22,131,264)(19,257,132,23,261,136)(20,129,262,24,133,258)(25,250,247,29,254,243)(26,244,255,30,248,251)(27,252,241,31,256,245)(28,246,249,32,242,253)(33,238,214,37,234,210)(34,211,235,38,215,239)(35,240,216,39,236,212)(36,213,237,40,209,233)(41,124,158,45,128,154)(42,155,121,46,159,125)(43,126,160,47,122,156)(44,157,123,48,153,127)(49,284,274,53,288,278)(50,279,281,54,275,285)(51,286,276,55,282,280)(52,273,283,56,277,287)(65,102,106,69,98,110)(66,111,99,70,107,103)(67,104,108,71,100,112)(68,105,101,72,109,97)(73,182,195,77,178,199)(74,200,179,78,196,183)(75,184,197,79,180,193)(76,194,181,80,198,177)(81,206,161,85,202,165)(82,166,203,86,162,207)(83,208,163,87,204,167)(84,168,205,88,164,201)(89,142,152,93,138,148)(90,149,139,94,145,143)(91,144,146,95,140,150)(92,151,141,96,147,137)(169,217,188,173,221,192)(170,185,222,174,189,218)(171,219,190,175,223,186)(172,187,224,176,191,220), (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)(225,226,227,228,229,230,231,232)(233,234,235,236,237,238,239,240)(241,242,243,244,245,246,247,248)(249,250,251,252,253,254,255,256)(257,258,259,260,261,262,263,264)(265,266,267,268,269,270,271,272)(273,274,275,276,277,278,279,280)(281,282,283,284,285,286,287,288), (1,123,170,20)(2,23,171,126)(3,121,172,18)(4,21,173,124)(5,127,174,24)(6,19,175,122)(7,125,176,22)(8,17,169,128)(9,89,179,248)(10,243,180,92)(11,95,181,246)(12,241,182,90)(13,93,183,244)(14,247,184,96)(15,91,177,242)(16,245,178,94)(25,79,151,228)(26,231,152,74)(27,77,145,226)(28,229,146,80)(29,75,147,232)(30,227,148,78)(31,73,149,230)(32,225,150,76)(33,287,86,104)(34,99,87,282)(35,285,88,102)(36,97,81,288)(37,283,82,100)(38,103,83,286)(39,281,84,98)(40,101,85,284)(41,120,130,221)(42,224,131,115)(43,118,132,219)(44,222,133,113)(45,116,134,217)(46,220,135,119)(47,114,136,223)(48,218,129,117)(49,209,105,202)(50,205,106,212)(51,215,107,208)(52,203,108,210)(53,213,109,206)(54,201,110,216)(55,211,111,204)(56,207,112,214)(57,155,187,264)(58,259,188,158)(59,153,189,262)(60,257,190,156)(61,159,191,260)(62,263,192,154)(63,157,185,258)(64,261,186,160)(65,240,275,164)(66,167,276,235)(67,238,277,162)(68,165,278,233)(69,236,279,168)(70,163,280,239)(71,234,273,166)(72,161,274,237)(137,269,254,193)(138,196,255,272)(139,267,256,199)(140,194,249,270)(141,265,250,197)(142,200,251,268)(143,271,252,195)(144,198,253,266)>;

G:=Group( (1,11,285,170,181,102)(2,103,182,171,286,12)(3,13,287,172,183,104)(4,97,184,173,288,14)(5,15,281,174,177,98)(6,99,178,175,282,16)(7,9,283,176,179,100)(8,101,180,169,284,10)(17,243,40,128,92,85)(18,86,93,121,33,244)(19,245,34,122,94,87)(20,88,95,123,35,246)(21,247,36,124,96,81)(22,82,89,125,37,248)(23,241,38,126,90,83)(24,84,91,127,39,242)(25,209,154,151,202,263)(26,264,203,152,155,210)(27,211,156,145,204,257)(28,258,205,146,157,212)(29,213,158,147,206,259)(30,260,207,148,159,214)(31,215,160,149,208,261)(32,262,201,150,153,216)(41,141,165,130,250,233)(42,234,251,131,166,142)(43,143,167,132,252,235)(44,236,253,133,168,144)(45,137,161,134,254,237)(46,238,255,135,162,138)(47,139,163,136,256,239)(48,240,249,129,164,140)(49,228,62,105,79,192)(50,185,80,106,63,229)(51,230,64,107,73,186)(52,187,74,108,57,231)(53,232,58,109,75,188)(54,189,76,110,59,225)(55,226,60,111,77,190)(56,191,78,112,61,227)(65,117,270,275,218,194)(66,195,219,276,271,118)(67,119,272,277,220,196)(68,197,221,278,265,120)(69,113,266,279,222,198)(70,199,223,280,267,114)(71,115,268,273,224,200)(72,193,217,274,269,116), (1,63,113,5,59,117)(2,118,60,6,114,64)(3,57,115,7,61,119)(4,120,62,8,116,58)(9,227,272,13,231,268)(10,269,232,14,265,228)(11,229,266,15,225,270)(12,271,226,16,267,230)(17,263,130,21,259,134)(18,135,260,22,131,264)(19,257,132,23,261,136)(20,129,262,24,133,258)(25,250,247,29,254,243)(26,244,255,30,248,251)(27,252,241,31,256,245)(28,246,249,32,242,253)(33,238,214,37,234,210)(34,211,235,38,215,239)(35,240,216,39,236,212)(36,213,237,40,209,233)(41,124,158,45,128,154)(42,155,121,46,159,125)(43,126,160,47,122,156)(44,157,123,48,153,127)(49,284,274,53,288,278)(50,279,281,54,275,285)(51,286,276,55,282,280)(52,273,283,56,277,287)(65,102,106,69,98,110)(66,111,99,70,107,103)(67,104,108,71,100,112)(68,105,101,72,109,97)(73,182,195,77,178,199)(74,200,179,78,196,183)(75,184,197,79,180,193)(76,194,181,80,198,177)(81,206,161,85,202,165)(82,166,203,86,162,207)(83,208,163,87,204,167)(84,168,205,88,164,201)(89,142,152,93,138,148)(90,149,139,94,145,143)(91,144,146,95,140,150)(92,151,141,96,147,137)(169,217,188,173,221,192)(170,185,222,174,189,218)(171,219,190,175,223,186)(172,187,224,176,191,220), (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)(225,226,227,228,229,230,231,232)(233,234,235,236,237,238,239,240)(241,242,243,244,245,246,247,248)(249,250,251,252,253,254,255,256)(257,258,259,260,261,262,263,264)(265,266,267,268,269,270,271,272)(273,274,275,276,277,278,279,280)(281,282,283,284,285,286,287,288), (1,123,170,20)(2,23,171,126)(3,121,172,18)(4,21,173,124)(5,127,174,24)(6,19,175,122)(7,125,176,22)(8,17,169,128)(9,89,179,248)(10,243,180,92)(11,95,181,246)(12,241,182,90)(13,93,183,244)(14,247,184,96)(15,91,177,242)(16,245,178,94)(25,79,151,228)(26,231,152,74)(27,77,145,226)(28,229,146,80)(29,75,147,232)(30,227,148,78)(31,73,149,230)(32,225,150,76)(33,287,86,104)(34,99,87,282)(35,285,88,102)(36,97,81,288)(37,283,82,100)(38,103,83,286)(39,281,84,98)(40,101,85,284)(41,120,130,221)(42,224,131,115)(43,118,132,219)(44,222,133,113)(45,116,134,217)(46,220,135,119)(47,114,136,223)(48,218,129,117)(49,209,105,202)(50,205,106,212)(51,215,107,208)(52,203,108,210)(53,213,109,206)(54,201,110,216)(55,211,111,204)(56,207,112,214)(57,155,187,264)(58,259,188,158)(59,153,189,262)(60,257,190,156)(61,159,191,260)(62,263,192,154)(63,157,185,258)(64,261,186,160)(65,240,275,164)(66,167,276,235)(67,238,277,162)(68,165,278,233)(69,236,279,168)(70,163,280,239)(71,234,273,166)(72,161,274,237)(137,269,254,193)(138,196,255,272)(139,267,256,199)(140,194,249,270)(141,265,250,197)(142,200,251,268)(143,271,252,195)(144,198,253,266) );

G=PermutationGroup([(1,11,285,170,181,102),(2,103,182,171,286,12),(3,13,287,172,183,104),(4,97,184,173,288,14),(5,15,281,174,177,98),(6,99,178,175,282,16),(7,9,283,176,179,100),(8,101,180,169,284,10),(17,243,40,128,92,85),(18,86,93,121,33,244),(19,245,34,122,94,87),(20,88,95,123,35,246),(21,247,36,124,96,81),(22,82,89,125,37,248),(23,241,38,126,90,83),(24,84,91,127,39,242),(25,209,154,151,202,263),(26,264,203,152,155,210),(27,211,156,145,204,257),(28,258,205,146,157,212),(29,213,158,147,206,259),(30,260,207,148,159,214),(31,215,160,149,208,261),(32,262,201,150,153,216),(41,141,165,130,250,233),(42,234,251,131,166,142),(43,143,167,132,252,235),(44,236,253,133,168,144),(45,137,161,134,254,237),(46,238,255,135,162,138),(47,139,163,136,256,239),(48,240,249,129,164,140),(49,228,62,105,79,192),(50,185,80,106,63,229),(51,230,64,107,73,186),(52,187,74,108,57,231),(53,232,58,109,75,188),(54,189,76,110,59,225),(55,226,60,111,77,190),(56,191,78,112,61,227),(65,117,270,275,218,194),(66,195,219,276,271,118),(67,119,272,277,220,196),(68,197,221,278,265,120),(69,113,266,279,222,198),(70,199,223,280,267,114),(71,115,268,273,224,200),(72,193,217,274,269,116)], [(1,63,113,5,59,117),(2,118,60,6,114,64),(3,57,115,7,61,119),(4,120,62,8,116,58),(9,227,272,13,231,268),(10,269,232,14,265,228),(11,229,266,15,225,270),(12,271,226,16,267,230),(17,263,130,21,259,134),(18,135,260,22,131,264),(19,257,132,23,261,136),(20,129,262,24,133,258),(25,250,247,29,254,243),(26,244,255,30,248,251),(27,252,241,31,256,245),(28,246,249,32,242,253),(33,238,214,37,234,210),(34,211,235,38,215,239),(35,240,216,39,236,212),(36,213,237,40,209,233),(41,124,158,45,128,154),(42,155,121,46,159,125),(43,126,160,47,122,156),(44,157,123,48,153,127),(49,284,274,53,288,278),(50,279,281,54,275,285),(51,286,276,55,282,280),(52,273,283,56,277,287),(65,102,106,69,98,110),(66,111,99,70,107,103),(67,104,108,71,100,112),(68,105,101,72,109,97),(73,182,195,77,178,199),(74,200,179,78,196,183),(75,184,197,79,180,193),(76,194,181,80,198,177),(81,206,161,85,202,165),(82,166,203,86,162,207),(83,208,163,87,204,167),(84,168,205,88,164,201),(89,142,152,93,138,148),(90,149,139,94,145,143),(91,144,146,95,140,150),(92,151,141,96,147,137),(169,217,188,173,221,192),(170,185,222,174,189,218),(171,219,190,175,223,186),(172,187,224,176,191,220)], [(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),(225,226,227,228,229,230,231,232),(233,234,235,236,237,238,239,240),(241,242,243,244,245,246,247,248),(249,250,251,252,253,254,255,256),(257,258,259,260,261,262,263,264),(265,266,267,268,269,270,271,272),(273,274,275,276,277,278,279,280),(281,282,283,284,285,286,287,288)], [(1,123,170,20),(2,23,171,126),(3,121,172,18),(4,21,173,124),(5,127,174,24),(6,19,175,122),(7,125,176,22),(8,17,169,128),(9,89,179,248),(10,243,180,92),(11,95,181,246),(12,241,182,90),(13,93,183,244),(14,247,184,96),(15,91,177,242),(16,245,178,94),(25,79,151,228),(26,231,152,74),(27,77,145,226),(28,229,146,80),(29,75,147,232),(30,227,148,78),(31,73,149,230),(32,225,150,76),(33,287,86,104),(34,99,87,282),(35,285,88,102),(36,97,81,288),(37,283,82,100),(38,103,83,286),(39,281,84,98),(40,101,85,284),(41,120,130,221),(42,224,131,115),(43,118,132,219),(44,222,133,113),(45,116,134,217),(46,220,135,119),(47,114,136,223),(48,218,129,117),(49,209,105,202),(50,205,106,212),(51,215,107,208),(52,203,108,210),(53,213,109,206),(54,201,110,216),(55,211,111,204),(56,207,112,214),(57,155,187,264),(58,259,188,158),(59,153,189,262),(60,257,190,156),(61,159,191,260),(62,263,192,154),(63,157,185,258),(64,261,186,160),(65,240,275,164),(66,167,276,235),(67,238,277,162),(68,165,278,233),(69,236,279,168),(70,163,280,239),(71,234,273,166),(72,161,274,237),(137,269,254,193),(138,196,255,272),(139,267,256,199),(140,194,249,270),(141,265,250,197),(142,200,251,268),(143,271,252,195),(144,198,253,266)])

54 conjugacy classes

class 1 2A2B2C3A3B3C3D4A4B4C4D4E4F6A···6L8A8B8C8D12A···12X
order122233334444446···6888812···12
size11112222224436362···2181818184···4

54 irreducible representations

dim1111122222222244
type++++++++--+-
imageC1C2C2C2C4S3D4D4D6Dic3SD16Q16C3⋊D4C3⋊D4Q82S3C3⋊Q16
kernelC62.117D4C2×C324C8C12⋊Dic3Q8×C3×C6Q8×C32C6×Q8C3×C12C62C2×C12C3×Q8C3×C6C3×C6C12C2×C6C6C6
# reps1111441148228844

Matrix representation of C62.117D4 in GL6(𝔽73)

0720000
110000
0072300
0072200
0000720
0000072
,
72720000
100000
001000
000100
0000720
0000072
,
50550000
5230000
00474000
00362600
00001657
00001616
,
48370000
62250000
00263300
00374700
00006243
00004311

G:=sub<GL(6,GF(73))| [0,1,0,0,0,0,72,1,0,0,0,0,0,0,72,72,0,0,0,0,3,2,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[72,1,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[50,5,0,0,0,0,55,23,0,0,0,0,0,0,47,36,0,0,0,0,40,26,0,0,0,0,0,0,16,16,0,0,0,0,57,16],[48,62,0,0,0,0,37,25,0,0,0,0,0,0,26,37,0,0,0,0,33,47,0,0,0,0,0,0,62,43,0,0,0,0,43,11] >;

C62.117D4 in GAP, Magma, Sage, TeX

C_6^2._{117}D_4
% in TeX

G:=Group("C6^2.117D4");
// GroupNames label

G:=SmallGroup(288,310);
// by ID

G=gap.SmallGroup(288,310);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,28,141,120,675,346,80,2693,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^6=b^6=1,c^4=b^3,d^2=a^3,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=a^3*c^3>;
// generators/relations

׿
×
𝔽