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G = C4×C68order 272 = 24·17

Abelian group of type [4,68]

direct product, abelian, monomial, 2-elementary

Aliases: C4×C68, SmallGroup(272,20)

Series: Derived Chief Lower central Upper central

C1 — C4×C68
C1C2C22C2×C34C2×C68 — C4×C68
C1 — C4×C68
C1 — C4×C68

Generators and relations for C4×C68
 G = < a,b | a4=b68=1, ab=ba >


Smallest permutation representation of C4×C68
Regular action on 272 points
Generators in S272
(1 186 213 98)(2 187 214 99)(3 188 215 100)(4 189 216 101)(5 190 217 102)(6 191 218 103)(7 192 219 104)(8 193 220 105)(9 194 221 106)(10 195 222 107)(11 196 223 108)(12 197 224 109)(13 198 225 110)(14 199 226 111)(15 200 227 112)(16 201 228 113)(17 202 229 114)(18 203 230 115)(19 204 231 116)(20 137 232 117)(21 138 233 118)(22 139 234 119)(23 140 235 120)(24 141 236 121)(25 142 237 122)(26 143 238 123)(27 144 239 124)(28 145 240 125)(29 146 241 126)(30 147 242 127)(31 148 243 128)(32 149 244 129)(33 150 245 130)(34 151 246 131)(35 152 247 132)(36 153 248 133)(37 154 249 134)(38 155 250 135)(39 156 251 136)(40 157 252 69)(41 158 253 70)(42 159 254 71)(43 160 255 72)(44 161 256 73)(45 162 257 74)(46 163 258 75)(47 164 259 76)(48 165 260 77)(49 166 261 78)(50 167 262 79)(51 168 263 80)(52 169 264 81)(53 170 265 82)(54 171 266 83)(55 172 267 84)(56 173 268 85)(57 174 269 86)(58 175 270 87)(59 176 271 88)(60 177 272 89)(61 178 205 90)(62 179 206 91)(63 180 207 92)(64 181 208 93)(65 182 209 94)(66 183 210 95)(67 184 211 96)(68 185 212 97)
(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)

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

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

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

272 conjugacy classes

class 1 2A2B2C4A···4L17A···17P34A···34AV68A···68GJ
order12224···417···1734···3468···68
size11111···11···11···11···1

272 irreducible representations

dim111111
type++
imageC1C2C4C17C34C68
kernelC4×C68C2×C68C68C42C2×C4C4
# reps13121648192

Matrix representation of C4×C68 in GL2(𝔽137) generated by

370
0100
,
1260
0133
G:=sub<GL(2,GF(137))| [37,0,0,100],[126,0,0,133] >;

C4×C68 in GAP, Magma, Sage, TeX

C_4\times C_{68}
% in TeX

G:=Group("C4xC68");
// GroupNames label

G:=SmallGroup(272,20);
// by ID

G=gap.SmallGroup(272,20);
# by ID

G:=PCGroup([5,-2,-2,-17,-2,-2,340,686]);
// Polycyclic

G:=Group<a,b|a^4=b^68=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C4×C68 in TeX

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