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G = C122.C2order 288 = 25·32

13rd non-split extension by C122 of C2 acting faithfully

metabelian, supersoluble, monomial

Aliases: C122.13C2, C12.78(C4×S3), (C4×C12).21S3, (C6×C12).18C4, C324C87C4, C326(C8⋊C4), (C2×C12).415D6, C42.1(C3⋊S3), (C3×C6).17C42, C6.13(C4×Dic3), C62.101(C2×C4), (C2×C12).11Dic3, (C3×C6).21M4(2), C6.8(C4.Dic3), (C6×C12).337C22, C32(C42.S3), C2.1(C12.58D6), C4.19(C4×C3⋊S3), C2.3(C4×C3⋊Dic3), (C3×C12).110(C2×C4), (C2×C4).2(C3⋊Dic3), (C2×C6).43(C2×Dic3), C22.7(C2×C3⋊Dic3), (C2×C324C8).15C2, (C2×C4).88(C2×C3⋊S3), SmallGroup(288,278)

Series: Derived Chief Lower central Upper central

C1C3×C6 — C122.C2
C1C3C32C3×C6C3×C12C6×C12C2×C324C8 — C122.C2
C32C3×C6 — C122.C2
C1C2×C4C42

Generators and relations for C122.C2
 G = < a,b,c | a12=b12=1, c2=b9, ab=ba, cac-1=a5b6, cbc-1=b5 >

Subgroups: 244 in 120 conjugacy classes, 73 normal (11 characteristic)
C1, C2, C2 [×2], C3 [×4], C4 [×2], C4 [×2], C22, C6 [×12], C8 [×4], C2×C4, C2×C4 [×2], C32, C12 [×8], C12 [×8], C2×C6 [×4], C42, C2×C8 [×2], C3×C6, C3×C6 [×2], C3⋊C8 [×16], C2×C12 [×12], C8⋊C4, C3×C12 [×2], C3×C12 [×2], C62, C2×C3⋊C8 [×8], C4×C12 [×4], C324C8 [×4], C6×C12, C6×C12 [×2], C42.S3 [×4], C2×C324C8 [×2], C122, C122.C2
Quotients: C1, C2 [×3], C4 [×6], C22, S3 [×4], C2×C4 [×3], Dic3 [×8], D6 [×4], C42, M4(2) [×2], C3⋊S3, C4×S3 [×8], C2×Dic3 [×4], C8⋊C4, C3⋊Dic3 [×2], C2×C3⋊S3, C4.Dic3 [×8], C4×Dic3 [×4], C4×C3⋊S3 [×2], C2×C3⋊Dic3, C42.S3 [×4], C12.58D6 [×2], C4×C3⋊Dic3, C122.C2

Smallest permutation representation of C122.C2
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)
(1 198 272 71 166 135 111 171 211 228 282 60)(2 199 273 72 167 136 112 172 212 217 283 49)(3 200 274 61 168 137 113 173 213 218 284 50)(4 201 275 62 157 138 114 174 214 219 285 51)(5 202 276 63 158 139 115 175 215 220 286 52)(6 203 265 64 159 140 116 176 216 221 287 53)(7 204 266 65 160 141 117 177 205 222 288 54)(8 193 267 66 161 142 118 178 206 223 277 55)(9 194 268 67 162 143 119 179 207 224 278 56)(10 195 269 68 163 144 120 180 208 225 279 57)(11 196 270 69 164 133 109 169 209 226 280 58)(12 197 271 70 165 134 110 170 210 227 281 59)(13 33 98 75 244 191 87 152 127 236 255 45)(14 34 99 76 245 192 88 153 128 237 256 46)(15 35 100 77 246 181 89 154 129 238 257 47)(16 36 101 78 247 182 90 155 130 239 258 48)(17 25 102 79 248 183 91 156 131 240 259 37)(18 26 103 80 249 184 92 145 132 229 260 38)(19 27 104 81 250 185 93 146 121 230 261 39)(20 28 105 82 251 186 94 147 122 231 262 40)(21 29 106 83 252 187 95 148 123 232 263 41)(22 30 107 84 241 188 96 149 124 233 264 42)(23 31 108 73 242 189 85 150 125 234 253 43)(24 32 97 74 243 190 86 151 126 235 254 44)
(1 15 228 238 111 89 71 77)(2 94 217 82 112 20 72 231)(3 13 218 236 113 87 61 75)(4 92 219 80 114 18 62 229)(5 23 220 234 115 85 63 73)(6 90 221 78 116 16 64 239)(7 21 222 232 117 95 65 83)(8 88 223 76 118 14 66 237)(9 19 224 230 119 93 67 81)(10 86 225 74 120 24 68 235)(11 17 226 240 109 91 69 79)(12 96 227 84 110 22 70 233)(25 270 259 58 156 209 248 133)(26 214 260 138 145 275 249 51)(27 268 261 56 146 207 250 143)(28 212 262 136 147 273 251 49)(29 266 263 54 148 205 252 141)(30 210 264 134 149 271 241 59)(31 276 253 52 150 215 242 139)(32 208 254 144 151 269 243 57)(33 274 255 50 152 213 244 137)(34 206 256 142 153 267 245 55)(35 272 257 60 154 211 246 135)(36 216 258 140 155 265 247 53)(37 164 131 196 183 280 102 169)(38 285 132 174 184 157 103 201)(39 162 121 194 185 278 104 179)(40 283 122 172 186 167 105 199)(41 160 123 204 187 288 106 177)(42 281 124 170 188 165 107 197)(43 158 125 202 189 286 108 175)(44 279 126 180 190 163 97 195)(45 168 127 200 191 284 98 173)(46 277 128 178 192 161 99 193)(47 166 129 198 181 282 100 171)(48 287 130 176 182 159 101 203)

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

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

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

84 conjugacy classes

class 1 2A2B2C3A3B3C3D4A4B4C4D4E4F4G4H6A···6L8A···8H12A···12AV
order12223333444444446···68···812···12
size11112222111122222···218···182···2

84 irreducible representations

dim11111222222
type++++-+
imageC1C2C2C4C4S3Dic3D6M4(2)C4×S3C4.Dic3
kernelC122.C2C2×C324C8C122C324C8C6×C12C4×C12C2×C12C2×C12C3×C6C12C6
# reps1218448441632

Matrix representation of C122.C2 in GL4(𝔽73) generated by

3000
02400
00700
00024
,
64000
0800
0030
00024
,
0100
1000
00028
00590
G:=sub<GL(4,GF(73))| [3,0,0,0,0,24,0,0,0,0,70,0,0,0,0,24],[64,0,0,0,0,8,0,0,0,0,3,0,0,0,0,24],[0,1,0,0,1,0,0,0,0,0,0,59,0,0,28,0] >;

C122.C2 in GAP, Magma, Sage, TeX

C_{12}^2.C_2
% in TeX

G:=Group("C12^2.C2");
// GroupNames label

G:=SmallGroup(288,278);
// by ID

G=gap.SmallGroup(288,278);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,28,253,64,100,2693,9414]);
// Polycyclic

G:=Group<a,b,c|a^12=b^12=1,c^2=b^9,a*b=b*a,c*a*c^-1=a^5*b^6,c*b*c^-1=b^5>;
// generators/relations

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