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## 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 [×3], C3, C4 [×4], C4 [×7], C22, C6 [×3], C2×C4 [×3], C2×C4 [×4], Q8 [×4], C9, Dic3 [×6], C12 [×4], C12, C2×C6, C42, C42 [×2], C4⋊C4 [×3], C2×Q8, C18 [×3], Dic6 [×4], C2×Dic3 [×4], C2×C12 [×3], C4×Q8, Dic9 [×4], Dic9 [×2], C36 [×4], C36, C2×C18, C4×Dic3 [×2], Dic3⋊C4 [×2], C4⋊Dic3, C4×C12, C2×Dic6, Dic18 [×4], C2×Dic9 [×4], C2×C36 [×3], C4×Dic6, C4×Dic9 [×2], Dic9⋊C4 [×2], C4⋊Dic9, C4×C36, C2×Dic18, C4×Dic18
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], Q8 [×2], C23, D6 [×3], C22×C4, C2×Q8, C4○D4, D9, Dic6 [×2], C4×S3 [×2], C22×S3, C4×Q8, D18 [×3], C2×Dic6, S3×C2×C4, C4○D12, Dic18 [×2], C4×D9 [×2], 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
(1 179 269 99)(2 180 270 100)(3 145 271 101)(4 146 272 102)(5 147 273 103)(6 148 274 104)(7 149 275 105)(8 150 276 106)(9 151 277 107)(10 152 278 108)(11 153 279 73)(12 154 280 74)(13 155 281 75)(14 156 282 76)(15 157 283 77)(16 158 284 78)(17 159 285 79)(18 160 286 80)(19 161 287 81)(20 162 288 82)(21 163 253 83)(22 164 254 84)(23 165 255 85)(24 166 256 86)(25 167 257 87)(26 168 258 88)(27 169 259 89)(28 170 260 90)(29 171 261 91)(30 172 262 92)(31 173 263 93)(32 174 264 94)(33 175 265 95)(34 176 266 96)(35 177 267 97)(36 178 268 98)(37 249 121 209)(38 250 122 210)(39 251 123 211)(40 252 124 212)(41 217 125 213)(42 218 126 214)(43 219 127 215)(44 220 128 216)(45 221 129 181)(46 222 130 182)(47 223 131 183)(48 224 132 184)(49 225 133 185)(50 226 134 186)(51 227 135 187)(52 228 136 188)(53 229 137 189)(54 230 138 190)(55 231 139 191)(56 232 140 192)(57 233 141 193)(58 234 142 194)(59 235 143 195)(60 236 144 196)(61 237 109 197)(62 238 110 198)(63 239 111 199)(64 240 112 200)(65 241 113 201)(66 242 114 202)(67 243 115 203)(68 244 116 204)(69 245 117 205)(70 246 118 206)(71 247 119 207)(72 248 120 208)
(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)
(1 118 19 136)(2 117 20 135)(3 116 21 134)(4 115 22 133)(5 114 23 132)(6 113 24 131)(7 112 25 130)(8 111 26 129)(9 110 27 128)(10 109 28 127)(11 144 29 126)(12 143 30 125)(13 142 31 124)(14 141 32 123)(15 140 33 122)(16 139 34 121)(17 138 35 120)(18 137 36 119)(37 284 55 266)(38 283 56 265)(39 282 57 264)(40 281 58 263)(41 280 59 262)(42 279 60 261)(43 278 61 260)(44 277 62 259)(45 276 63 258)(46 275 64 257)(47 274 65 256)(48 273 66 255)(49 272 67 254)(50 271 68 253)(51 270 69 288)(52 269 70 287)(53 268 71 286)(54 267 72 285)(73 236 91 218)(74 235 92 217)(75 234 93 252)(76 233 94 251)(77 232 95 250)(78 231 96 249)(79 230 97 248)(80 229 98 247)(81 228 99 246)(82 227 100 245)(83 226 101 244)(84 225 102 243)(85 224 103 242)(86 223 104 241)(87 222 105 240)(88 221 106 239)(89 220 107 238)(90 219 108 237)(145 204 163 186)(146 203 164 185)(147 202 165 184)(148 201 166 183)(149 200 167 182)(150 199 168 181)(151 198 169 216)(152 197 170 215)(153 196 171 214)(154 195 172 213)(155 194 173 212)(156 193 174 211)(157 192 175 210)(158 191 176 209)(159 190 177 208)(160 189 178 207)(161 188 179 206)(162 187 180 205)

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

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

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

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