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G = C72⋊C4order 288 = 25·32

4th semidirect product of C72 and C4 acting via C4/C2=C2

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C724C4, C83Dic9, C18.4C42, C24.9Dic3, C18.2M4(2), C9⋊C84C4, (C2×C8).8D9, C92(C8⋊C4), C4.21(C4×D9), C3.(C24⋊C4), C12.70(C4×S3), (C2×C24).26S3, C36.26(C2×C4), (C2×C72).12C2, (C2×C4).92D18, C2.4(C4×Dic9), C6.9(C4×Dic3), C6.5(C8⋊S3), (C2×C12).406D6, C2.2(C8⋊D9), (C4×Dic9).6C2, (C2×Dic9).3C4, C4.12(C2×Dic9), C22.10(C4×D9), C12.52(C2×Dic3), (C2×C36).104C22, (C2×C9⋊C8).10C2, (C2×C6).36(C4×S3), (C2×C18).11(C2×C4), SmallGroup(288,23)

Series: Derived Chief Lower central Upper central

C1C18 — C72⋊C4
C1C3C9C18C2×C18C2×C36C4×Dic9 — C72⋊C4
C9C18 — C72⋊C4
C1C2×C4C2×C8

Generators and relations for C72⋊C4
 G = < a,b | a72=b4=1, bab-1=a53 >

18C4
18C4
9C8
9C2×C4
9C8
9C2×C4
6Dic3
6Dic3
9C2×C8
9C42
3C3⋊C8
3C3⋊C8
3C2×Dic3
3C2×Dic3
2Dic9
2Dic9
9C8⋊C4
3C4×Dic3
3C2×C3⋊C8
3C24⋊C4

Smallest permutation representation of C72⋊C4
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)
(1 193 226 138)(2 174 227 119)(3 155 228 100)(4 208 229 81)(5 189 230 134)(6 170 231 115)(7 151 232 96)(8 204 233 77)(9 185 234 130)(10 166 235 111)(11 147 236 92)(12 200 237 73)(13 181 238 126)(14 162 239 107)(15 215 240 88)(16 196 241 141)(17 177 242 122)(18 158 243 103)(19 211 244 84)(20 192 245 137)(21 173 246 118)(22 154 247 99)(23 207 248 80)(24 188 249 133)(25 169 250 114)(26 150 251 95)(27 203 252 76)(28 184 253 129)(29 165 254 110)(30 146 255 91)(31 199 256 144)(32 180 257 125)(33 161 258 106)(34 214 259 87)(35 195 260 140)(36 176 261 121)(37 157 262 102)(38 210 263 83)(39 191 264 136)(40 172 265 117)(41 153 266 98)(42 206 267 79)(43 187 268 132)(44 168 269 113)(45 149 270 94)(46 202 271 75)(47 183 272 128)(48 164 273 109)(49 145 274 90)(50 198 275 143)(51 179 276 124)(52 160 277 105)(53 213 278 86)(54 194 279 139)(55 175 280 120)(56 156 281 101)(57 209 282 82)(58 190 283 135)(59 171 284 116)(60 152 285 97)(61 205 286 78)(62 186 287 131)(63 167 288 112)(64 148 217 93)(65 201 218 74)(66 182 219 127)(67 163 220 108)(68 216 221 89)(69 197 222 142)(70 178 223 123)(71 159 224 104)(72 212 225 85)

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

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

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

84 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D4E4F4G4H6A6B6C8A8B8C8D8E8F8G8H9A9B9C12A12B12C12D18A···18I24A···24H36A···36L72A···72X
order1222344444444666888888889991212121218···1824···2436···3672···72
size1111211111818181822222221818181822222222···22···22···22···2

84 irreducible representations

dim11111112222222222222
type+++++-++-+
imageC1C2C2C2C4C4C4S3Dic3D6M4(2)D9C4×S3C4×S3Dic9D18C8⋊S3C4×D9C4×D9C8⋊D9
kernelC72⋊C4C2×C9⋊C8C4×Dic9C2×C72C9⋊C8C72C2×Dic9C2×C24C24C2×C12C18C2×C8C12C2×C6C8C2×C4C6C4C22C2
# reps111144412143226386624

Matrix representation of C72⋊C4 in GL3(𝔽73) generated by

2700
03920
05319
,
2700
06341
05110
G:=sub<GL(3,GF(73))| [27,0,0,0,39,53,0,20,19],[27,0,0,0,63,51,0,41,10] >;

C72⋊C4 in GAP, Magma, Sage, TeX

C_{72}\rtimes C_4
% in TeX

G:=Group("C72:C4");
// GroupNames label

G:=SmallGroup(288,23);
// by ID

G=gap.SmallGroup(288,23);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,28,253,64,80,6725,292,9414]);
// Polycyclic

G:=Group<a,b|a^72=b^4=1,b*a*b^-1=a^53>;
// generators/relations

Export

Subgroup lattice of C72⋊C4 in TeX

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