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## G = C4.Dic18order 288 = 25·32

### 2nd non-split extension by C4 of Dic18 acting via Dic18/Dic9=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C36 — C4.Dic18
 Chief series C1 — C3 — C9 — C18 — C2×C18 — C2×C36 — C2×C9⋊C8 — C4.Dic18
 Lower central C9 — C18 — C36 — C4.Dic18
 Upper central C1 — C22 — C2×C4 — C4⋊C4

Generators and relations for C4.Dic18
G = < a,b,c | a4=b36=1, c2=ab18, bab-1=a-1, ac=ca, cbc-1=a-1b-1 >

Smallest permutation representation of C4.Dic18
Regular action on 288 points
Generators in S288
(1 227 273 40)(2 41 274 228)(3 229 275 42)(4 43 276 230)(5 231 277 44)(6 45 278 232)(7 233 279 46)(8 47 280 234)(9 235 281 48)(10 49 282 236)(11 237 283 50)(12 51 284 238)(13 239 285 52)(14 53 286 240)(15 241 287 54)(16 55 288 242)(17 243 253 56)(18 57 254 244)(19 245 255 58)(20 59 256 246)(21 247 257 60)(22 61 258 248)(23 249 259 62)(24 63 260 250)(25 251 261 64)(26 65 262 252)(27 217 263 66)(28 67 264 218)(29 219 265 68)(30 69 266 220)(31 221 267 70)(32 71 268 222)(33 223 269 72)(34 37 270 224)(35 225 271 38)(36 39 272 226)(73 159 197 111)(74 112 198 160)(75 161 199 113)(76 114 200 162)(77 163 201 115)(78 116 202 164)(79 165 203 117)(80 118 204 166)(81 167 205 119)(82 120 206 168)(83 169 207 121)(84 122 208 170)(85 171 209 123)(86 124 210 172)(87 173 211 125)(88 126 212 174)(89 175 213 127)(90 128 214 176)(91 177 215 129)(92 130 216 178)(93 179 181 131)(94 132 182 180)(95 145 183 133)(96 134 184 146)(97 147 185 135)(98 136 186 148)(99 149 187 137)(100 138 188 150)(101 151 189 139)(102 140 190 152)(103 153 191 141)(104 142 192 154)(105 155 193 143)(106 144 194 156)(107 157 195 109)(108 110 196 158)
(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 88 245 144 273 212 58 156)(2 125 59 105 274 173 246 193)(3 86 247 142 275 210 60 154)(4 123 61 103 276 171 248 191)(5 84 249 140 277 208 62 152)(6 121 63 101 278 169 250 189)(7 82 251 138 279 206 64 150)(8 119 65 99 280 167 252 187)(9 80 217 136 281 204 66 148)(10 117 67 97 282 165 218 185)(11 78 219 134 283 202 68 146)(12 115 69 95 284 163 220 183)(13 76 221 132 285 200 70 180)(14 113 71 93 286 161 222 181)(15 74 223 130 287 198 72 178)(16 111 37 91 288 159 224 215)(17 108 225 128 253 196 38 176)(18 109 39 89 254 157 226 213)(19 106 227 126 255 194 40 174)(20 143 41 87 256 155 228 211)(21 104 229 124 257 192 42 172)(22 141 43 85 258 153 230 209)(23 102 231 122 259 190 44 170)(24 139 45 83 260 151 232 207)(25 100 233 120 261 188 46 168)(26 137 47 81 262 149 234 205)(27 98 235 118 263 186 48 166)(28 135 49 79 264 147 236 203)(29 96 237 116 265 184 50 164)(30 133 51 77 266 145 238 201)(31 94 239 114 267 182 52 162)(32 131 53 75 268 179 240 199)(33 92 241 112 269 216 54 160)(34 129 55 73 270 177 242 197)(35 90 243 110 271 214 56 158)(36 127 57 107 272 175 244 195)

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

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

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

54 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 4F 6A 6B 6C 8A 8B 8C 8D 9A 9B 9C 12A ··· 12F 18A ··· 18I 36A ··· 36R order 1 2 2 2 3 4 4 4 4 4 4 6 6 6 8 8 8 8 9 9 9 12 ··· 12 18 ··· 18 36 ··· 36 size 1 1 1 1 2 2 2 4 4 36 36 2 2 2 18 18 18 18 2 2 2 4 ··· 4 2 ··· 2 4 ··· 4

54 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + - + + + - + - - + - + image C1 C2 C2 C2 C4 S3 Q8 D4 D6 SD16 D9 Dic6 C4×S3 C3⋊D4 D18 Dic18 C4×D9 C9⋊D4 D4.S3 Q8⋊2S3 D4.D9 Q8⋊2D9 kernel C4.Dic18 C2×C9⋊C8 C4⋊Dic9 C9×C4⋊C4 C9⋊C8 C3×C4⋊C4 C36 C2×C18 C2×C12 C18 C4⋊C4 C12 C12 C2×C6 C2×C4 C4 C4 C22 C6 C6 C2 C2 # reps 1 1 1 1 4 1 1 1 1 4 3 2 2 2 3 6 6 6 1 1 3 3

Matrix representation of C4.Dic18 in GL4(𝔽73) generated by

 1 0 0 0 0 1 0 0 0 0 0 72 0 0 1 0
,
 54 25 0 0 48 29 0 0 0 0 7 13 0 0 13 66
,
 14 19 0 0 5 59 0 0 0 0 6 67 0 0 6 6
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,0,1,0,0,72,0],[54,48,0,0,25,29,0,0,0,0,7,13,0,0,13,66],[14,5,0,0,19,59,0,0,0,0,6,6,0,0,67,6] >;

C4.Dic18 in GAP, Magma, Sage, TeX

C_4.{\rm Dic}_{18}
% in TeX

G:=Group("C4.Dic18");
// GroupNames label

G:=SmallGroup(288,15);
// by ID

G=gap.SmallGroup(288,15);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,56,365,36,346,80,6725,292,9414]);
// Polycyclic

G:=Group<a,b,c|a^4=b^36=1,c^2=a*b^18,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=a^-1*b^-1>;
// generators/relations

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