Copied to
clipboard

G = C22×C60order 240 = 24·3·5

Abelian group of type [2,2,60]

direct product, abelian, monomial, 2-elementary

Aliases: C22×C60, SmallGroup(240,185)

Series: Derived Chief Lower central Upper central

C1 — C22×C60
C1C2C10C30C60C2×C60 — C22×C60
C1 — C22×C60
C1 — C22×C60

Generators and relations for C22×C60
 G = < a,b,c | a2=b2=c60=1, ab=ba, ac=ca, bc=cb >

Subgroups: 108, all normal (16 characteristic)
C1, C2, C2 [×6], C3, C4 [×4], C22 [×7], C5, C6, C6 [×6], C2×C4 [×6], C23, C10, C10 [×6], C12 [×4], C2×C6 [×7], C15, C22×C4, C20 [×4], C2×C10 [×7], C2×C12 [×6], C22×C6, C30, C30 [×6], C2×C20 [×6], C22×C10, C22×C12, C60 [×4], C2×C30 [×7], C22×C20, C2×C60 [×6], C22×C30, C22×C60
Quotients: C1, C2 [×7], C3, C4 [×4], C22 [×7], C5, C6 [×7], C2×C4 [×6], C23, C10 [×7], C12 [×4], C2×C6 [×7], C15, C22×C4, C20 [×4], C2×C10 [×7], C2×C12 [×6], C22×C6, C30 [×7], C2×C20 [×6], C22×C10, C22×C12, C60 [×4], C2×C30 [×7], C22×C20, C2×C60 [×6], C22×C30, C22×C60

Smallest permutation representation of C22×C60
Regular action on 240 points
Generators in S240
(1 138)(2 139)(3 140)(4 141)(5 142)(6 143)(7 144)(8 145)(9 146)(10 147)(11 148)(12 149)(13 150)(14 151)(15 152)(16 153)(17 154)(18 155)(19 156)(20 157)(21 158)(22 159)(23 160)(24 161)(25 162)(26 163)(27 164)(28 165)(29 166)(30 167)(31 168)(32 169)(33 170)(34 171)(35 172)(36 173)(37 174)(38 175)(39 176)(40 177)(41 178)(42 179)(43 180)(44 121)(45 122)(46 123)(47 124)(48 125)(49 126)(50 127)(51 128)(52 129)(53 130)(54 131)(55 132)(56 133)(57 134)(58 135)(59 136)(60 137)(61 233)(62 234)(63 235)(64 236)(65 237)(66 238)(67 239)(68 240)(69 181)(70 182)(71 183)(72 184)(73 185)(74 186)(75 187)(76 188)(77 189)(78 190)(79 191)(80 192)(81 193)(82 194)(83 195)(84 196)(85 197)(86 198)(87 199)(88 200)(89 201)(90 202)(91 203)(92 204)(93 205)(94 206)(95 207)(96 208)(97 209)(98 210)(99 211)(100 212)(101 213)(102 214)(103 215)(104 216)(105 217)(106 218)(107 219)(108 220)(109 221)(110 222)(111 223)(112 224)(113 225)(114 226)(115 227)(116 228)(117 229)(118 230)(119 231)(120 232)
(1 213)(2 214)(3 215)(4 216)(5 217)(6 218)(7 219)(8 220)(9 221)(10 222)(11 223)(12 224)(13 225)(14 226)(15 227)(16 228)(17 229)(18 230)(19 231)(20 232)(21 233)(22 234)(23 235)(24 236)(25 237)(26 238)(27 239)(28 240)(29 181)(30 182)(31 183)(32 184)(33 185)(34 186)(35 187)(36 188)(37 189)(38 190)(39 191)(40 192)(41 193)(42 194)(43 195)(44 196)(45 197)(46 198)(47 199)(48 200)(49 201)(50 202)(51 203)(52 204)(53 205)(54 206)(55 207)(56 208)(57 209)(58 210)(59 211)(60 212)(61 158)(62 159)(63 160)(64 161)(65 162)(66 163)(67 164)(68 165)(69 166)(70 167)(71 168)(72 169)(73 170)(74 171)(75 172)(76 173)(77 174)(78 175)(79 176)(80 177)(81 178)(82 179)(83 180)(84 121)(85 122)(86 123)(87 124)(88 125)(89 126)(90 127)(91 128)(92 129)(93 130)(94 131)(95 132)(96 133)(97 134)(98 135)(99 136)(100 137)(101 138)(102 139)(103 140)(104 141)(105 142)(106 143)(107 144)(108 145)(109 146)(110 147)(111 148)(112 149)(113 150)(114 151)(115 152)(116 153)(117 154)(118 155)(119 156)(120 157)
(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)

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

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

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

C22×C60 is a maximal subgroup of   C60.212D4  C30.29C42  C60.205D4  C23.26D30  C23.28D30  C6029D4

240 conjugacy classes

class 1 2A···2G3A3B4A···4H5A5B5C5D6A···6N10A···10AB12A···12P15A···15H20A···20AF30A···30BD60A···60BL
order12···2334···455556···610···1012···1215···1520···2030···3060···60
size11···1111···111111···11···11···11···11···11···11···1

240 irreducible representations

dim1111111111111111
type+++
imageC1C2C2C3C4C5C6C6C10C10C12C15C20C30C30C60
kernelC22×C60C2×C60C22×C30C22×C20C2×C30C22×C12C2×C20C22×C10C2×C12C22×C6C2×C10C22×C4C2×C6C2×C4C23C22
# reps1612841222441683248864

Matrix representation of C22×C60 in GL3(𝔽61) generated by

6000
010
001
,
6000
0600
001
,
3500
0160
006
G:=sub<GL(3,GF(61))| [60,0,0,0,1,0,0,0,1],[60,0,0,0,60,0,0,0,1],[35,0,0,0,16,0,0,0,6] >;

C22×C60 in GAP, Magma, Sage, TeX

C_2^2\times C_{60}
% in TeX

G:=Group("C2^2xC60");
// GroupNames label

G:=SmallGroup(240,185);
// by ID

G=gap.SmallGroup(240,185);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-5,-2,720]);
// Polycyclic

G:=Group<a,b,c|a^2=b^2=c^60=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations

׿
×
𝔽