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## G = C122.C2order 288 = 25·32

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C122.C2
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C6×C12 — C2×C32⋊4C8 — C122.C2
 Lower central C32 — C3×C6 — C122.C2
 Upper central C1 — C2×C4 — C42

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, C3, C4, C4, C22, C6, C8, C2×C4, C2×C4, C32, C12, C12, C2×C6, C42, C2×C8, C3×C6, C3×C6, C3⋊C8, C2×C12, C8⋊C4, C3×C12, C3×C12, C62, C2×C3⋊C8, C4×C12, C324C8, C6×C12, C6×C12, C42.S3, C2×C324C8, C122, C122.C2
Quotients: C1, C2, C4, C22, S3, C2×C4, Dic3, D6, C42, M4(2), C3⋊S3, C4×S3, C2×Dic3, C8⋊C4, C3⋊Dic3, C2×C3⋊S3, C4.Dic3, C4×Dic3, C4×C3⋊S3, C2×C3⋊Dic3, C42.S3, C12.58D6, 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 28 69 268 38 199 231 107 176 254 220 77)(2 29 70 269 39 200 232 108 177 255 221 78)(3 30 71 270 40 201 233 97 178 256 222 79)(4 31 72 271 41 202 234 98 179 257 223 80)(5 32 61 272 42 203 235 99 180 258 224 81)(6 33 62 273 43 204 236 100 169 259 225 82)(7 34 63 274 44 193 237 101 170 260 226 83)(8 35 64 275 45 194 238 102 171 261 227 84)(9 36 65 276 46 195 239 103 172 262 228 73)(10 25 66 265 47 196 240 104 173 263 217 74)(11 26 67 266 48 197 229 105 174 264 218 75)(12 27 68 267 37 198 230 106 175 253 219 76)(13 208 86 121 279 192 116 162 150 139 56 247)(14 209 87 122 280 181 117 163 151 140 57 248)(15 210 88 123 281 182 118 164 152 141 58 249)(16 211 89 124 282 183 119 165 153 142 59 250)(17 212 90 125 283 184 120 166 154 143 60 251)(18 213 91 126 284 185 109 167 155 144 49 252)(19 214 92 127 285 186 110 168 156 133 50 241)(20 215 93 128 286 187 111 157 145 134 51 242)(21 216 94 129 287 188 112 158 146 135 52 243)(22 205 95 130 288 189 113 159 147 136 53 244)(23 206 96 131 277 190 114 160 148 137 54 245)(24 207 85 132 278 191 115 161 149 138 55 246)
(1 126 254 18 231 144 268 109)(2 137 255 114 232 131 269 23)(3 124 256 16 233 142 270 119)(4 135 257 112 234 129 271 21)(5 122 258 14 235 140 272 117)(6 133 259 110 236 127 273 19)(7 132 260 24 237 138 274 115)(8 143 261 120 238 125 275 17)(9 130 262 22 239 136 276 113)(10 141 263 118 240 123 265 15)(11 128 264 20 229 134 266 111)(12 139 253 116 230 121 267 13)(25 88 217 249 104 152 47 182)(26 145 218 187 105 93 48 242)(27 86 219 247 106 150 37 192)(28 155 220 185 107 91 38 252)(29 96 221 245 108 148 39 190)(30 153 222 183 97 89 40 250)(31 94 223 243 98 146 41 188)(32 151 224 181 99 87 42 248)(33 92 225 241 100 156 43 186)(34 149 226 191 101 85 44 246)(35 90 227 251 102 154 45 184)(36 147 228 189 103 95 46 244)(49 176 167 199 284 69 213 77)(50 62 168 82 285 169 214 204)(51 174 157 197 286 67 215 75)(52 72 158 80 287 179 216 202)(53 172 159 195 288 65 205 73)(54 70 160 78 277 177 206 200)(55 170 161 193 278 63 207 83)(56 68 162 76 279 175 208 198)(57 180 163 203 280 61 209 81)(58 66 164 74 281 173 210 196)(59 178 165 201 282 71 211 79)(60 64 166 84 283 171 212 194)```

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

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

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

84 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 4A 4B 4C 4D 4E 4F 4G 4H 6A ··· 6L 8A ··· 8H 12A ··· 12AV order 1 2 2 2 3 3 3 3 4 4 4 4 4 4 4 4 6 ··· 6 8 ··· 8 12 ··· 12 size 1 1 1 1 2 2 2 2 1 1 1 1 2 2 2 2 2 ··· 2 18 ··· 18 2 ··· 2

84 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 type + + + + - + image C1 C2 C2 C4 C4 S3 Dic3 D6 M4(2) C4×S3 C4.Dic3 kernel C122.C2 C2×C32⋊4C8 C122 C32⋊4C8 C6×C12 C4×C12 C2×C12 C2×C12 C3×C6 C12 C6 # reps 1 2 1 8 4 4 8 4 4 16 32

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

 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 28 0 0 59 0
`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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