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G = C2×Dic30order 240 = 24·3·5

Direct product of C2 and Dic30

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C2×Dic30, C302Q8, C62Dic10, C102Dic6, C4.11D30, C20.46D6, C12.46D10, C22.8D30, C30.28C23, C60.53C22, Dic15.7C22, C155(C2×Q8), C53(C2×Dic6), (C2×C60).6C2, (C2×C20).4S3, (C2×C4).4D15, (C2×C12).4D5, C33(C2×Dic10), (C2×C10).26D6, (C2×C6).26D10, C6.28(C22×D5), C2.3(C22×D15), C10.28(C22×S3), (C2×C30).27C22, (C2×Dic15).3C2, SmallGroup(240,175)

Series: Derived Chief Lower central Upper central

C1C30 — C2×Dic30
C1C5C15C30Dic15C2×Dic15 — C2×Dic30
C15C30 — C2×Dic30
C1C22C2×C4

Generators and relations for C2×Dic30
 G = < a,b,c | a2=b60=1, c2=b30, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 296 in 76 conjugacy classes, 43 normal (17 characteristic)
C1, C2, C2 [×2], C3, C4 [×2], C4 [×4], C22, C5, C6, C6 [×2], C2×C4, C2×C4 [×2], Q8 [×4], C10, C10 [×2], Dic3 [×4], C12 [×2], C2×C6, C15, C2×Q8, Dic5 [×4], C20 [×2], C2×C10, Dic6 [×4], C2×Dic3 [×2], C2×C12, C30, C30 [×2], Dic10 [×4], C2×Dic5 [×2], C2×C20, C2×Dic6, Dic15 [×4], C60 [×2], C2×C30, C2×Dic10, Dic30 [×4], C2×Dic15 [×2], C2×C60, C2×Dic30
Quotients: C1, C2 [×7], C22 [×7], S3, Q8 [×2], C23, D5, D6 [×3], C2×Q8, D10 [×3], Dic6 [×2], C22×S3, D15, Dic10 [×2], C22×D5, C2×Dic6, D30 [×3], C2×Dic10, Dic30 [×2], C22×D15, C2×Dic30

Smallest permutation representation of C2×Dic30
Regular action on 240 points
Generators in S240
(1 153)(2 154)(3 155)(4 156)(5 157)(6 158)(7 159)(8 160)(9 161)(10 162)(11 163)(12 164)(13 165)(14 166)(15 167)(16 168)(17 169)(18 170)(19 171)(20 172)(21 173)(22 174)(23 175)(24 176)(25 177)(26 178)(27 179)(28 180)(29 121)(30 122)(31 123)(32 124)(33 125)(34 126)(35 127)(36 128)(37 129)(38 130)(39 131)(40 132)(41 133)(42 134)(43 135)(44 136)(45 137)(46 138)(47 139)(48 140)(49 141)(50 142)(51 143)(52 144)(53 145)(54 146)(55 147)(56 148)(57 149)(58 150)(59 151)(60 152)(61 230)(62 231)(63 232)(64 233)(65 234)(66 235)(67 236)(68 237)(69 238)(70 239)(71 240)(72 181)(73 182)(74 183)(75 184)(76 185)(77 186)(78 187)(79 188)(80 189)(81 190)(82 191)(83 192)(84 193)(85 194)(86 195)(87 196)(88 197)(89 198)(90 199)(91 200)(92 201)(93 202)(94 203)(95 204)(96 205)(97 206)(98 207)(99 208)(100 209)(101 210)(102 211)(103 212)(104 213)(105 214)(106 215)(107 216)(108 217)(109 218)(110 219)(111 220)(112 221)(113 222)(114 223)(115 224)(116 225)(117 226)(118 227)(119 228)(120 229)
(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)
(1 218 31 188)(2 217 32 187)(3 216 33 186)(4 215 34 185)(5 214 35 184)(6 213 36 183)(7 212 37 182)(8 211 38 181)(9 210 39 240)(10 209 40 239)(11 208 41 238)(12 207 42 237)(13 206 43 236)(14 205 44 235)(15 204 45 234)(16 203 46 233)(17 202 47 232)(18 201 48 231)(19 200 49 230)(20 199 50 229)(21 198 51 228)(22 197 52 227)(23 196 53 226)(24 195 54 225)(25 194 55 224)(26 193 56 223)(27 192 57 222)(28 191 58 221)(29 190 59 220)(30 189 60 219)(61 171 91 141)(62 170 92 140)(63 169 93 139)(64 168 94 138)(65 167 95 137)(66 166 96 136)(67 165 97 135)(68 164 98 134)(69 163 99 133)(70 162 100 132)(71 161 101 131)(72 160 102 130)(73 159 103 129)(74 158 104 128)(75 157 105 127)(76 156 106 126)(77 155 107 125)(78 154 108 124)(79 153 109 123)(80 152 110 122)(81 151 111 121)(82 150 112 180)(83 149 113 179)(84 148 114 178)(85 147 115 177)(86 146 116 176)(87 145 117 175)(88 144 118 174)(89 143 119 173)(90 142 120 172)

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

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

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

C2×Dic30 is a maximal subgroup of
C60.31D4  Dic3012C4  Dic3015C4  Dic309C4  Dic308C4  C4.D60  C60.63D4  D12.33D10  Dic3⋊Dic10  Dic3017C4  Dic3014C4  C60.68D4  C60.44D4  C60.45D4  C60.69D4  Dic5⋊Dic6  C60.48D4  D104Dic6  D63Dic10  Dic15.10D4  C608Q8  C427D15  C222Dic30  C23.11D30  Dic1510Q8  C4⋊Dic30  D305Q8  D306Q8  C8.D30  C60.205D4  C60.17D4  Dic154Q8  D4.9D30  C2×D5×Dic6  D20.39D6  C2×S3×Dic10  C2×Q8×D15  D4.10D30
C2×Dic30 is a maximal quotient of
C608Q8  C60.24Q8  C222Dic30  C4⋊Dic30  C4.Dic30  C60.205D4

66 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D4E4F5A5B6A6B6C10A···10F12A12B12C12D15A15B15C15D20A···20H30A···30L60A···60P
order122234444445566610···10121212121515151520···2030···3060···60
size111122230303030222222···2222222222···22···22···2

66 irreducible representations

dim11112222222222222
type+++++-+++++-+-++-
imageC1C2C2C2S3Q8D5D6D6D10D10Dic6D15Dic10D30D30Dic30
kernelC2×Dic30Dic30C2×Dic15C2×C60C2×C20C30C2×C12C20C2×C10C12C2×C6C10C2×C4C6C4C22C2
# reps142112221424488416

Matrix representation of C2×Dic30 in GL4(𝔽61) generated by

60000
06000
0010
0001
,
06000
11800
005512
004934
,
181800
604300
00110
005750
G:=sub<GL(4,GF(61))| [60,0,0,0,0,60,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,60,18,0,0,0,0,55,49,0,0,12,34],[18,60,0,0,18,43,0,0,0,0,11,57,0,0,0,50] >;

C2×Dic30 in GAP, Magma, Sage, TeX

C_2\times {\rm Dic}_{30}
% in TeX

G:=Group("C2xDic30");
// GroupNames label

G:=SmallGroup(240,175);
// by ID

G=gap.SmallGroup(240,175);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-5,48,218,50,964,6917]);
// Polycyclic

G:=Group<a,b,c|a^2=b^60=1,c^2=b^30,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

׿
×
𝔽