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## G = C4×C72order 288 = 25·32

### Abelian group of type [4,72]

Aliases: C4×C72, SmallGroup(288,46)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4×C72
 Chief series C1 — C2 — C6 — C2×C6 — C2×C12 — C2×C36 — C2×C72 — C4×C72
 Lower central C1 — C4×C72
 Upper central 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, C3, C4, C22, C6, C6, C8, C2×C4, C2×C4, C9, C12, C2×C6, C42, C2×C8, C18, C18, C24, C2×C12, C2×C12, C4×C8, C36, C2×C18, C4×C12, C2×C24, C72, C2×C36, C2×C36, C4×C24, C4×C36, C2×C72, C4×C72
Quotients: C1, C2, C3, C4, C22, C6, C8, C2×C4, C9, C12, C2×C6, C42, C2×C8, C18, C24, C2×C12, C4×C8, C36, C2×C18, C4×C12, C2×C24, C72, C2×C36, C4×C24, C4×C36, C2×C72, C4×C72

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

288 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 type + + + image C1 C2 C2 C3 C4 C4 C6 C6 C8 C9 C12 C12 C18 C18 C24 C36 C36 C72 kernel C4×C72 C4×C36 C2×C72 C4×C24 C72 C2×C36 C4×C12 C2×C24 C36 C4×C8 C24 C2×C12 C42 C2×C8 C12 C8 C2×C4 C4 # reps 1 1 2 2 8 4 2 4 16 6 16 8 6 12 32 48 24 96

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

 27 0 0 1
,
 71 0 0 33
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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