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G = C2×C122order 244 = 22·61

Abelian group of type [2,122]

direct product, abelian, monomial, 2-elementary

Aliases: C2×C122, SmallGroup(244,5)

Series: Derived Chief Lower central Upper central

C1 — C2×C122
C1C61C122 — C2×C122
C1 — C2×C122
C1 — C2×C122

Generators and relations for C2×C122
 G = < a,b | a2=b122=1, ab=ba >


Smallest permutation representation of C2×C122
Regular action on 244 points
Generators in S244
(1 225)(2 226)(3 227)(4 228)(5 229)(6 230)(7 231)(8 232)(9 233)(10 234)(11 235)(12 236)(13 237)(14 238)(15 239)(16 240)(17 241)(18 242)(19 243)(20 244)(21 123)(22 124)(23 125)(24 126)(25 127)(26 128)(27 129)(28 130)(29 131)(30 132)(31 133)(32 134)(33 135)(34 136)(35 137)(36 138)(37 139)(38 140)(39 141)(40 142)(41 143)(42 144)(43 145)(44 146)(45 147)(46 148)(47 149)(48 150)(49 151)(50 152)(51 153)(52 154)(53 155)(54 156)(55 157)(56 158)(57 159)(58 160)(59 161)(60 162)(61 163)(62 164)(63 165)(64 166)(65 167)(66 168)(67 169)(68 170)(69 171)(70 172)(71 173)(72 174)(73 175)(74 176)(75 177)(76 178)(77 179)(78 180)(79 181)(80 182)(81 183)(82 184)(83 185)(84 186)(85 187)(86 188)(87 189)(88 190)(89 191)(90 192)(91 193)(92 194)(93 195)(94 196)(95 197)(96 198)(97 199)(98 200)(99 201)(100 202)(101 203)(102 204)(103 205)(104 206)(105 207)(106 208)(107 209)(108 210)(109 211)(110 212)(111 213)(112 214)(113 215)(114 216)(115 217)(116 218)(117 219)(118 220)(119 221)(120 222)(121 223)(122 224)
(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)

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

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

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

C2×C122 is a maximal subgroup of   C61⋊D4

244 conjugacy classes

class 1 2A2B2C61A···61BH122A···122FX
order122261···61122···122
size11111···11···1

244 irreducible representations

dim1111
type++
imageC1C2C61C122
kernelC2×C122C122C22C2
# reps1360180

Matrix representation of C2×C122 in GL2(𝔽367) generated by

3660
01
,
750
0230
G:=sub<GL(2,GF(367))| [366,0,0,1],[75,0,0,230] >;

C2×C122 in GAP, Magma, Sage, TeX

C_2\times C_{122}
% in TeX

G:=Group("C2xC122");
// GroupNames label

G:=SmallGroup(244,5);
// by ID

G=gap.SmallGroup(244,5);
# by ID

G:=PCGroup([3,-2,-2,-61]);
// Polycyclic

G:=Group<a,b|a^2=b^122=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C2×C122 in TeX

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