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G = C4×C72order 288 = 25·32

Abelian group of type [4,72]

direct product, abelian, monomial, 2-elementary

Aliases: C4×C72, SmallGroup(288,46)

Series: Derived Chief Lower central Upper central

C1 — C4×C72
C1C2C6C2×C6C2×C12C2×C36C2×C72 — C4×C72
C1 — C4×C72
C1 — C4×C72

Generators and relations for C4×C72
 G = < a,b | a4=b72=1, ab=ba >

Subgroups: 66, all normal (18 characteristic)
C1, C2, C2 [×2], C3, C4 [×6], C22, C6, C6 [×2], C8 [×4], C2×C4, C2×C4 [×2], C9, C12 [×6], C2×C6, C42, C2×C8 [×2], C18, C18 [×2], C24 [×4], C2×C12, C2×C12 [×2], C4×C8, C36 [×6], C2×C18, C4×C12, C2×C24 [×2], C72 [×4], C2×C36, C2×C36 [×2], C4×C24, C4×C36, C2×C72 [×2], C4×C72
Quotients: C1, C2 [×3], C3, C4 [×6], C22, C6 [×3], C8 [×4], C2×C4 [×3], C9, C12 [×6], C2×C6, C42, C2×C8 [×2], C18 [×3], C24 [×4], C2×C12 [×3], C4×C8, C36 [×6], C2×C18, C4×C12, C2×C24 [×2], C72 [×4], C2×C36 [×3], C4×C24, C4×C36, C2×C72 [×2], C4×C72

Smallest permutation representation of C4×C72
Regular action on 288 points
Generators in S288
(1 269 82 193)(2 270 83 194)(3 271 84 195)(4 272 85 196)(5 273 86 197)(6 274 87 198)(7 275 88 199)(8 276 89 200)(9 277 90 201)(10 278 91 202)(11 279 92 203)(12 280 93 204)(13 281 94 205)(14 282 95 206)(15 283 96 207)(16 284 97 208)(17 285 98 209)(18 286 99 210)(19 287 100 211)(20 288 101 212)(21 217 102 213)(22 218 103 214)(23 219 104 215)(24 220 105 216)(25 221 106 145)(26 222 107 146)(27 223 108 147)(28 224 109 148)(29 225 110 149)(30 226 111 150)(31 227 112 151)(32 228 113 152)(33 229 114 153)(34 230 115 154)(35 231 116 155)(36 232 117 156)(37 233 118 157)(38 234 119 158)(39 235 120 159)(40 236 121 160)(41 237 122 161)(42 238 123 162)(43 239 124 163)(44 240 125 164)(45 241 126 165)(46 242 127 166)(47 243 128 167)(48 244 129 168)(49 245 130 169)(50 246 131 170)(51 247 132 171)(52 248 133 172)(53 249 134 173)(54 250 135 174)(55 251 136 175)(56 252 137 176)(57 253 138 177)(58 254 139 178)(59 255 140 179)(60 256 141 180)(61 257 142 181)(62 258 143 182)(63 259 144 183)(64 260 73 184)(65 261 74 185)(66 262 75 186)(67 263 76 187)(68 264 77 188)(69 265 78 189)(70 266 79 190)(71 267 80 191)(72 268 81 192)
(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)

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

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

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

288 conjugacy classes

class 1 2A2B2C3A3B4A···4L6A···6F8A···8P9A···9F12A···12X18A···18R24A···24AF36A···36BT72A···72CR
order1222334···46···68···89···912···1218···1824···2436···3672···72
size1111111···11···11···11···11···11···11···11···11···1

288 irreducible representations

dim111111111111111111
type+++
imageC1C2C2C3C4C4C6C6C8C9C12C12C18C18C24C36C36C72
kernelC4×C72C4×C36C2×C72C4×C24C72C2×C36C4×C12C2×C24C36C4×C8C24C2×C12C42C2×C8C12C8C2×C4C4
# reps1122842416616861232482496

Matrix representation of C4×C72 in GL2(𝔽73) generated by

270
01
,
710
033
G:=sub<GL(2,GF(73))| [27,0,0,1],[71,0,0,33] >;

C4×C72 in GAP, Magma, Sage, TeX

C_4\times C_{72}
% in TeX

G:=Group("C4xC72");
// GroupNames label

G:=SmallGroup(288,46);
// by ID

G=gap.SmallGroup(288,46);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-2,84,176,268,360]);
// Polycyclic

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

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