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G = C3×C96order 288 = 25·32

Abelian group of type [3,96]

direct product, abelian, monomial, 3-elementary

Aliases: C3×C96, SmallGroup(288,66)

Series: Derived Chief Lower central Upper central

C1 — C3×C96
C1C2C4C8C16C48C3×C48 — C3×C96
C1 — C3×C96
C1 — C3×C96

Generators and relations for C3×C96
 G = < a,b | a3=b96=1, ab=ba >


Smallest permutation representation of C3×C96
Regular action on 288 points
Generators in S288
(1 280 138)(2 281 139)(3 282 140)(4 283 141)(5 284 142)(6 285 143)(7 286 144)(8 287 145)(9 288 146)(10 193 147)(11 194 148)(12 195 149)(13 196 150)(14 197 151)(15 198 152)(16 199 153)(17 200 154)(18 201 155)(19 202 156)(20 203 157)(21 204 158)(22 205 159)(23 206 160)(24 207 161)(25 208 162)(26 209 163)(27 210 164)(28 211 165)(29 212 166)(30 213 167)(31 214 168)(32 215 169)(33 216 170)(34 217 171)(35 218 172)(36 219 173)(37 220 174)(38 221 175)(39 222 176)(40 223 177)(41 224 178)(42 225 179)(43 226 180)(44 227 181)(45 228 182)(46 229 183)(47 230 184)(48 231 185)(49 232 186)(50 233 187)(51 234 188)(52 235 189)(53 236 190)(54 237 191)(55 238 192)(56 239 97)(57 240 98)(58 241 99)(59 242 100)(60 243 101)(61 244 102)(62 245 103)(63 246 104)(64 247 105)(65 248 106)(66 249 107)(67 250 108)(68 251 109)(69 252 110)(70 253 111)(71 254 112)(72 255 113)(73 256 114)(74 257 115)(75 258 116)(76 259 117)(77 260 118)(78 261 119)(79 262 120)(80 263 121)(81 264 122)(82 265 123)(83 266 124)(84 267 125)(85 268 126)(86 269 127)(87 270 128)(88 271 129)(89 272 130)(90 273 131)(91 274 132)(92 275 133)(93 276 134)(94 277 135)(95 278 136)(96 279 137)
(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)| (1,280,138)(2,281,139)(3,282,140)(4,283,141)(5,284,142)(6,285,143)(7,286,144)(8,287,145)(9,288,146)(10,193,147)(11,194,148)(12,195,149)(13,196,150)(14,197,151)(15,198,152)(16,199,153)(17,200,154)(18,201,155)(19,202,156)(20,203,157)(21,204,158)(22,205,159)(23,206,160)(24,207,161)(25,208,162)(26,209,163)(27,210,164)(28,211,165)(29,212,166)(30,213,167)(31,214,168)(32,215,169)(33,216,170)(34,217,171)(35,218,172)(36,219,173)(37,220,174)(38,221,175)(39,222,176)(40,223,177)(41,224,178)(42,225,179)(43,226,180)(44,227,181)(45,228,182)(46,229,183)(47,230,184)(48,231,185)(49,232,186)(50,233,187)(51,234,188)(52,235,189)(53,236,190)(54,237,191)(55,238,192)(56,239,97)(57,240,98)(58,241,99)(59,242,100)(60,243,101)(61,244,102)(62,245,103)(63,246,104)(64,247,105)(65,248,106)(66,249,107)(67,250,108)(68,251,109)(69,252,110)(70,253,111)(71,254,112)(72,255,113)(73,256,114)(74,257,115)(75,258,116)(76,259,117)(77,260,118)(78,261,119)(79,262,120)(80,263,121)(81,264,122)(82,265,123)(83,266,124)(84,267,125)(85,268,126)(86,269,127)(87,270,128)(88,271,129)(89,272,130)(90,273,131)(91,274,132)(92,275,133)(93,276,134)(94,277,135)(95,278,136)(96,279,137), (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( (1,280,138)(2,281,139)(3,282,140)(4,283,141)(5,284,142)(6,285,143)(7,286,144)(8,287,145)(9,288,146)(10,193,147)(11,194,148)(12,195,149)(13,196,150)(14,197,151)(15,198,152)(16,199,153)(17,200,154)(18,201,155)(19,202,156)(20,203,157)(21,204,158)(22,205,159)(23,206,160)(24,207,161)(25,208,162)(26,209,163)(27,210,164)(28,211,165)(29,212,166)(30,213,167)(31,214,168)(32,215,169)(33,216,170)(34,217,171)(35,218,172)(36,219,173)(37,220,174)(38,221,175)(39,222,176)(40,223,177)(41,224,178)(42,225,179)(43,226,180)(44,227,181)(45,228,182)(46,229,183)(47,230,184)(48,231,185)(49,232,186)(50,233,187)(51,234,188)(52,235,189)(53,236,190)(54,237,191)(55,238,192)(56,239,97)(57,240,98)(58,241,99)(59,242,100)(60,243,101)(61,244,102)(62,245,103)(63,246,104)(64,247,105)(65,248,106)(66,249,107)(67,250,108)(68,251,109)(69,252,110)(70,253,111)(71,254,112)(72,255,113)(73,256,114)(74,257,115)(75,258,116)(76,259,117)(77,260,118)(78,261,119)(79,262,120)(80,263,121)(81,264,122)(82,265,123)(83,266,124)(84,267,125)(85,268,126)(86,269,127)(87,270,128)(88,271,129)(89,272,130)(90,273,131)(91,274,132)(92,275,133)(93,276,134)(94,277,135)(95,278,136)(96,279,137), (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([(1,280,138),(2,281,139),(3,282,140),(4,283,141),(5,284,142),(6,285,143),(7,286,144),(8,287,145),(9,288,146),(10,193,147),(11,194,148),(12,195,149),(13,196,150),(14,197,151),(15,198,152),(16,199,153),(17,200,154),(18,201,155),(19,202,156),(20,203,157),(21,204,158),(22,205,159),(23,206,160),(24,207,161),(25,208,162),(26,209,163),(27,210,164),(28,211,165),(29,212,166),(30,213,167),(31,214,168),(32,215,169),(33,216,170),(34,217,171),(35,218,172),(36,219,173),(37,220,174),(38,221,175),(39,222,176),(40,223,177),(41,224,178),(42,225,179),(43,226,180),(44,227,181),(45,228,182),(46,229,183),(47,230,184),(48,231,185),(49,232,186),(50,233,187),(51,234,188),(52,235,189),(53,236,190),(54,237,191),(55,238,192),(56,239,97),(57,240,98),(58,241,99),(59,242,100),(60,243,101),(61,244,102),(62,245,103),(63,246,104),(64,247,105),(65,248,106),(66,249,107),(67,250,108),(68,251,109),(69,252,110),(70,253,111),(71,254,112),(72,255,113),(73,256,114),(74,257,115),(75,258,116),(76,259,117),(77,260,118),(78,261,119),(79,262,120),(80,263,121),(81,264,122),(82,265,123),(83,266,124),(84,267,125),(85,268,126),(86,269,127),(87,270,128),(88,271,129),(89,272,130),(90,273,131),(91,274,132),(92,275,133),(93,276,134),(94,277,135),(95,278,136),(96,279,137)], [(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  2 3A···3H4A4B6A···6H8A8B8C8D12A···12P16A···16H24A···24AF32A···32P48A···48BL96A···96DX
order123···3446···6888812···1216···1624···2432···3248···4896···96
size111···1111···111111···11···11···11···11···11···1

288 irreducible representations

dim111111111111
type++
imageC1C2C3C4C6C8C12C16C24C32C48C96
kernelC3×C96C3×C48C96C3×C24C48C3×C12C24C3×C6C12C32C6C3
# reps118284168321664128

Matrix representation of C3×C96 in GL2(𝔽97) generated by

350
035
,
820
029
G:=sub<GL(2,GF(97))| [35,0,0,35],[82,0,0,29] >;

C3×C96 in GAP, Magma, Sage, TeX

C_3\times C_{96}
% in TeX

G:=Group("C3xC96");
// GroupNames label

G:=SmallGroup(288,66);
// by ID

G=gap.SmallGroup(288,66);
# by ID

G:=PCGroup([7,-2,-3,-3,-2,-2,-2,-2,126,80,102,124]);
// Polycyclic

G:=Group<a,b|a^3=b^96=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C3×C96 in TeX

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