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G = C721C4order 288 = 25·32

1st semidirect product of C72 and C4 acting via C4/C2=C2

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C721C4, C81Dic9, C18.4D8, C6.4D24, C2.1D72, C36.5Q8, C18.2Q16, C24.2Dic3, C2.2Dic36, C4.5Dic18, C6.2Dic12, C22.9D36, C12.20Dic6, (C2×C8).3D9, C92(C2.D8), (C2×C24).9S3, (C2×C72).5C2, C18.7(C4⋊C4), C3.(C241C4), C36.37(C2×C4), (C2×C4).75D18, (C2×C18).14D4, (C2×C6).22D12, C4⋊Dic9.3C2, C4.8(C2×Dic9), (C2×C12).364D6, C2.5(C4⋊Dic9), C6.9(C4⋊Dic3), (C2×C36).83C22, C12.43(C2×Dic3), SmallGroup(288,26)

Series: Derived Chief Lower central Upper central

C1C36 — C721C4
C1C3C9C18C2×C18C2×C36C4⋊Dic9 — C721C4
C9C18C36 — C721C4
C1C22C2×C4C2×C8

Generators and relations for C721C4
 G = < a,b | a72=b4=1, bab-1=a-1 >

36C4
36C4
18C2×C4
18C2×C4
12Dic3
12Dic3
9C4⋊C4
9C4⋊C4
6C2×Dic3
6C2×Dic3
4Dic9
4Dic9
9C2.D8
3C4⋊Dic3
3C4⋊Dic3
2C2×Dic9
2C2×Dic9
3C241C4

Smallest permutation representation of C721C4
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)
(1 231 121 162)(2 230 122 161)(3 229 123 160)(4 228 124 159)(5 227 125 158)(6 226 126 157)(7 225 127 156)(8 224 128 155)(9 223 129 154)(10 222 130 153)(11 221 131 152)(12 220 132 151)(13 219 133 150)(14 218 134 149)(15 217 135 148)(16 288 136 147)(17 287 137 146)(18 286 138 145)(19 285 139 216)(20 284 140 215)(21 283 141 214)(22 282 142 213)(23 281 143 212)(24 280 144 211)(25 279 73 210)(26 278 74 209)(27 277 75 208)(28 276 76 207)(29 275 77 206)(30 274 78 205)(31 273 79 204)(32 272 80 203)(33 271 81 202)(34 270 82 201)(35 269 83 200)(36 268 84 199)(37 267 85 198)(38 266 86 197)(39 265 87 196)(40 264 88 195)(41 263 89 194)(42 262 90 193)(43 261 91 192)(44 260 92 191)(45 259 93 190)(46 258 94 189)(47 257 95 188)(48 256 96 187)(49 255 97 186)(50 254 98 185)(51 253 99 184)(52 252 100 183)(53 251 101 182)(54 250 102 181)(55 249 103 180)(56 248 104 179)(57 247 105 178)(58 246 106 177)(59 245 107 176)(60 244 108 175)(61 243 109 174)(62 242 110 173)(63 241 111 172)(64 240 112 171)(65 239 113 170)(66 238 114 169)(67 237 115 168)(68 236 116 167)(69 235 117 166)(70 234 118 165)(71 233 119 164)(72 232 120 163)

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

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

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

78 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D4E4F6A6B6C8A8B8C8D9A9B9C12A12B12C12D18A···18I24A···24H36A···36L72A···72X
order1222344444466688889991212121218···1824···2436···3672···72
size111122236363636222222222222222···22···22···22···2

78 irreducible representations

dim1111222222222222222222
type++++-+-++-+-+-++--+-+
imageC1C2C2C4S3Q8D4Dic3D6D8Q16D9Dic6D12Dic9D18D24Dic12Dic18D36Dic36D72
kernelC721C4C4⋊Dic9C2×C72C72C2×C24C36C2×C18C24C2×C12C18C18C2×C8C12C2×C6C8C2×C4C6C6C4C22C2C2
# reps121411121223226344661212

Matrix representation of C721C4 in GL3(𝔽73) generated by

100
01113
06071
,
4600
01868
05055
G:=sub<GL(3,GF(73))| [1,0,0,0,11,60,0,13,71],[46,0,0,0,18,50,0,68,55] >;

C721C4 in GAP, Magma, Sage, TeX

C_{72}\rtimes_1C_4
% in TeX

G:=Group("C72:1C4");
// GroupNames label

G:=SmallGroup(288,26);
// by ID

G=gap.SmallGroup(288,26);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,28,141,176,675,80,6725,292,9414]);
// Polycyclic

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

Export

Subgroup lattice of C721C4 in TeX

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