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G = C122Dic6order 288 = 25·32

2nd semidirect product of C12 and Dic6 acting via Dic6/C6=C22

metabelian, supersoluble, monomial

Aliases: C122Dic6, C62.232C23, (C3×C12)⋊6Q8, C34(C12⋊Q8), C3⋊Dic38Q8, C6.43(S3×Q8), C6.112(S3×D4), (C2×C12).31D6, C3212(C4⋊Q8), C3⋊Dic3.47D4, C6.42(C2×Dic6), C41(C324Q8), (C6×C12).134C22, C6.Dic6.2C2, C12⋊Dic3.18C2, C2.4(Q8×C3⋊S3), C2.11(D4×C3⋊S3), C4⋊C4.4(C3⋊S3), (C3×C4⋊C4).21S3, (C3×C6).56(C2×Q8), (C3×C6).234(C2×D4), (C4×C3⋊Dic3).5C2, C2.7(C2×C324Q8), (C32×C4⋊C4).12C2, (C2×C324Q8).5C2, (C2×C6).249(C22×S3), C22.46(C22×C3⋊S3), (C2×C3⋊Dic3).157C22, (C2×C4).8(C2×C3⋊S3), SmallGroup(288,745)

Series: Derived Chief Lower central Upper central

C1C62 — C122Dic6
C1C3C32C3×C6C62C2×C3⋊Dic3C4×C3⋊Dic3 — C122Dic6
C32C62 — C122Dic6
C1C22C4⋊C4

Generators and relations for C122Dic6
 G = < a,b,c | a12=b12=1, c2=b6, bab-1=a7, cac-1=a5, cbc-1=b-1 >

Subgroups: 684 in 204 conjugacy classes, 81 normal (19 characteristic)
C1, C2 [×3], C3 [×4], C4 [×2], C4 [×8], C22, C6 [×12], C2×C4, C2×C4 [×2], C2×C4 [×4], Q8 [×4], C32, Dic3 [×24], C12 [×8], C12 [×8], C2×C6 [×4], C42, C4⋊C4, C4⋊C4 [×3], C2×Q8 [×2], C3×C6 [×3], Dic6 [×16], C2×Dic3 [×16], C2×C12 [×12], C4⋊Q8, C3⋊Dic3 [×4], C3⋊Dic3 [×2], C3×C12 [×2], C3×C12 [×2], C62, C4×Dic3 [×4], Dic3⋊C4 [×8], C4⋊Dic3 [×4], C3×C4⋊C4 [×4], C2×Dic6 [×8], C324Q8 [×4], C2×C3⋊Dic3 [×2], C2×C3⋊Dic3 [×2], C6×C12, C6×C12 [×2], C12⋊Q8 [×4], C4×C3⋊Dic3, C6.Dic6 [×2], C12⋊Dic3, C32×C4⋊C4, C2×C324Q8 [×2], C122Dic6
Quotients: C1, C2 [×7], C22 [×7], S3 [×4], D4 [×2], Q8 [×4], C23, D6 [×12], C2×D4, C2×Q8 [×2], C3⋊S3, Dic6 [×8], C22×S3 [×4], C4⋊Q8, C2×C3⋊S3 [×3], C2×Dic6 [×4], S3×D4 [×4], S3×Q8 [×4], C324Q8 [×2], C22×C3⋊S3, C12⋊Q8 [×4], C2×C324Q8, D4×C3⋊S3, Q8×C3⋊S3, C122Dic6

Smallest permutation representation of C122Dic6
Regular action on 288 points
Generators in S288
(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)
(1 153 52 70 32 137 89 101 17 40 131 113)(2 148 53 65 33 144 90 108 18 47 132 120)(3 155 54 72 34 139 91 103 19 42 121 115)(4 150 55 67 35 134 92 98 20 37 122 110)(5 145 56 62 36 141 93 105 21 44 123 117)(6 152 57 69 25 136 94 100 22 39 124 112)(7 147 58 64 26 143 95 107 23 46 125 119)(8 154 59 71 27 138 96 102 24 41 126 114)(9 149 60 66 28 133 85 97 13 48 127 109)(10 156 49 61 29 140 86 104 14 43 128 116)(11 151 50 68 30 135 87 99 15 38 129 111)(12 146 51 63 31 142 88 106 16 45 130 118)(73 253 227 265 185 157 247 235 283 169 215 197)(74 260 228 272 186 164 248 230 284 176 216 204)(75 255 217 267 187 159 249 237 285 171 205 199)(76 262 218 274 188 166 250 232 286 178 206 194)(77 257 219 269 189 161 251 239 287 173 207 201)(78 264 220 276 190 168 252 234 288 180 208 196)(79 259 221 271 191 163 241 229 277 175 209 203)(80 254 222 266 192 158 242 236 278 170 210 198)(81 261 223 273 181 165 243 231 279 177 211 193)(82 256 224 268 182 160 244 238 280 172 212 200)(83 263 225 275 183 167 245 233 281 179 213 195)(84 258 226 270 184 162 246 240 282 174 214 202)
(1 229 89 259)(2 234 90 264)(3 239 91 257)(4 232 92 262)(5 237 93 255)(6 230 94 260)(7 235 95 253)(8 240 96 258)(9 233 85 263)(10 238 86 256)(11 231 87 261)(12 236 88 254)(13 195 60 167)(14 200 49 160)(15 193 50 165)(16 198 51 158)(17 203 52 163)(18 196 53 168)(19 201 54 161)(20 194 55 166)(21 199 56 159)(22 204 57 164)(23 197 58 157)(24 202 59 162)(25 272 124 176)(26 265 125 169)(27 270 126 174)(28 275 127 179)(29 268 128 172)(30 273 129 177)(31 266 130 170)(32 271 131 175)(33 276 132 180)(34 269 121 173)(35 274 122 178)(36 267 123 171)(37 206 67 188)(38 211 68 181)(39 216 69 186)(40 209 70 191)(41 214 71 184)(42 207 72 189)(43 212 61 182)(44 205 62 187)(45 210 63 192)(46 215 64 185)(47 208 65 190)(48 213 66 183)(73 147 247 107)(74 152 248 100)(75 145 249 105)(76 150 250 98)(77 155 251 103)(78 148 252 108)(79 153 241 101)(80 146 242 106)(81 151 243 99)(82 156 244 104)(83 149 245 97)(84 154 246 102)(109 281 133 225)(110 286 134 218)(111 279 135 223)(112 284 136 228)(113 277 137 221)(114 282 138 226)(115 287 139 219)(116 280 140 224)(117 285 141 217)(118 278 142 222)(119 283 143 227)(120 288 144 220)

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

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

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

54 conjugacy classes

class 1 2A2B2C3A3B3C3D4A4B4C4D4E4F4G4H4I4J6A···6L12A···12X
order1222333344444444446···612···12
size1111222222441818181836362···24···4

54 irreducible representations

dim11111122222244
type++++++++--+-+-
imageC1C2C2C2C2C2S3D4Q8Q8D6Dic6S3×D4S3×Q8
kernelC122Dic6C4×C3⋊Dic3C6.Dic6C12⋊Dic3C32×C4⋊C4C2×C324Q8C3×C4⋊C4C3⋊Dic3C3⋊Dic3C3×C12C2×C12C12C6C6
# reps1121124222121644

Matrix representation of C122Dic6 in GL6(𝔽13)

1000000
240000
003900
0091000
0000120
0000012
,
600000
7110000
000100
0012000
000077
000002
,
820000
050000
0010400
004300
0000710
000086

G:=sub<GL(6,GF(13))| [10,2,0,0,0,0,0,4,0,0,0,0,0,0,3,9,0,0,0,0,9,10,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[6,7,0,0,0,0,0,11,0,0,0,0,0,0,0,12,0,0,0,0,1,0,0,0,0,0,0,0,7,0,0,0,0,0,7,2],[8,0,0,0,0,0,2,5,0,0,0,0,0,0,10,4,0,0,0,0,4,3,0,0,0,0,0,0,7,8,0,0,0,0,10,6] >;

C122Dic6 in GAP, Magma, Sage, TeX

C_{12}\rtimes_2{\rm Dic}_6
% in TeX

G:=Group("C12:2Dic6");
// GroupNames label

G:=SmallGroup(288,745);
// by ID

G=gap.SmallGroup(288,745);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,56,120,254,219,58,2693,9414]);
// Polycyclic

G:=Group<a,b,c|a^12=b^12=1,c^2=b^6,b*a*b^-1=a^7,c*a*c^-1=a^5,c*b*c^-1=b^-1>;
// generators/relations

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