Copied to
clipboard

## G = C4×Dic18order 288 = 25·32

### Direct product of C4 and Dic18

Series: Derived Chief Lower central Upper central

 Derived series C1 — C18 — C4×Dic18
 Chief series C1 — C3 — C9 — C18 — C2×C18 — C2×Dic9 — C2×Dic18 — C4×Dic18
 Lower central C9 — C18 — C4×Dic18
 Upper central C1 — C2×C4 — C42

Generators and relations for C4×Dic18
G = < a,b,c | a4=b36=1, c2=b18, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 344 in 105 conjugacy classes, 58 normal (30 characteristic)
C1, C2, C3, C4, C4, C22, C6, C2×C4, C2×C4, Q8, C9, Dic3, C12, C12, C2×C6, C42, C42, C4⋊C4, C2×Q8, C18, Dic6, C2×Dic3, C2×C12, C4×Q8, Dic9, Dic9, C36, C36, C2×C18, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C4×C12, C2×Dic6, Dic18, C2×Dic9, C2×C36, C4×Dic6, C4×Dic9, Dic9⋊C4, C4⋊Dic9, C4×C36, C2×Dic18, C4×Dic18
Quotients: C1, C2, C4, C22, S3, C2×C4, Q8, C23, D6, C22×C4, C2×Q8, C4○D4, D9, Dic6, C4×S3, C22×S3, C4×Q8, D18, C2×Dic6, S3×C2×C4, C4○D12, Dic18, C4×D9, C22×D9, C4×Dic6, C2×Dic18, C2×C4×D9, D365C2, C4×Dic18

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

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

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

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

84 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 4F 4G 4H 4I ··· 4P 6A 6B 6C 9A 9B 9C 12A ··· 12L 18A ··· 18I 36A ··· 36AJ order 1 2 2 2 3 4 4 4 4 4 4 4 4 4 ··· 4 6 6 6 9 9 9 12 ··· 12 18 ··· 18 36 ··· 36 size 1 1 1 1 2 1 1 1 1 2 2 2 2 18 ··· 18 2 2 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2

84 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + - + + - + - image C1 C2 C2 C2 C2 C2 C4 S3 Q8 D6 C4○D4 D9 Dic6 C4×S3 D18 C4○D12 Dic18 C4×D9 D36⋊5C2 kernel C4×Dic18 C4×Dic9 Dic9⋊C4 C4⋊Dic9 C4×C36 C2×Dic18 Dic18 C4×C12 C36 C2×C12 C18 C42 C12 C12 C2×C4 C6 C4 C4 C2 # reps 1 2 2 1 1 1 8 1 2 3 2 3 4 4 9 4 12 12 12

Matrix representation of C4×Dic18 in GL3(𝔽37) generated by

 31 0 0 0 36 0 0 0 36
,
 36 0 0 0 20 0 0 0 13
,
 1 0 0 0 0 1 0 36 0
G:=sub<GL(3,GF(37))| [31,0,0,0,36,0,0,0,36],[36,0,0,0,20,0,0,0,13],[1,0,0,0,0,36,0,1,0] >;

C4×Dic18 in GAP, Magma, Sage, TeX

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

G:=Group("C4xDic18");
// GroupNames label

G:=SmallGroup(288,78);
// by ID

G=gap.SmallGroup(288,78);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,253,120,58,6725,292,9414]);
// Polycyclic

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

׿
×
𝔽