Copied to
clipboard

G = D6011C4order 480 = 25·3·5

5th semidirect product of D60 and C4 acting via C4/C2=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D6011C4, Dic1520D4, C4⋊C48D15, C41(C4×D15), C126(C4×D5), C1537(C4×D4), C2010(C4×S3), C6012(C2×C4), C2.4(D4×D15), D3020(C2×C4), (C2×D60).8C2, (C2×C4).30D30, C6.104(D4×D5), C33(D208C4), C54(Dic35D4), (C4×Dic15)⋊3C2, (C2×C20).211D6, C10.106(S3×D4), C30.312(C2×D4), D303C412C2, (C2×C12).209D10, C30.258(C4○D4), C2.2(Q83D15), (C2×C60).178C22, C30.162(C22×C4), (C2×C30).290C23, C6.40(Q82D5), C10.40(Q83S3), C22.18(C22×D15), (C22×D15).82C22, (C2×Dic15).239C22, (C5×C4⋊C4)⋊4S3, (C3×C4⋊C4)⋊4D5, (C15×C4⋊C4)⋊4C2, C6.67(C2×C4×D5), (C2×C4×D15)⋊18C2, C10.99(S3×C2×C4), C2.13(C2×C4×D15), (C2×C6).286(C22×D5), (C2×C10).285(C22×S3), SmallGroup(480,858)

Series: Derived Chief Lower central Upper central

C1C30 — D6011C4
C1C5C15C30C2×C30C22×D15C2×D60 — D6011C4
C15C30 — D6011C4
C1C22C4⋊C4

Generators and relations for D6011C4
 G = < a,b,c | a60=b2=c4=1, bab=a-1, cac-1=a31, cbc-1=a30b >

Subgroups: 1188 in 188 conjugacy classes, 65 normal (33 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×2], C4 [×5], C22, C22 [×8], C5, S3 [×4], C6 [×3], C2×C4, C2×C4 [×2], C2×C4 [×6], D4 [×4], C23 [×2], D5 [×4], C10 [×3], Dic3 [×3], C12 [×2], C12 [×2], D6 [×8], C2×C6, C15, C42, C22⋊C4 [×2], C4⋊C4, C22×C4 [×2], C2×D4, Dic5 [×3], C20 [×2], C20 [×2], D10 [×8], C2×C10, C4×S3 [×4], D12 [×4], C2×Dic3 [×2], C2×C12, C2×C12 [×2], C22×S3 [×2], D15 [×4], C30 [×3], C4×D4, C4×D5 [×4], D20 [×4], C2×Dic5 [×2], C2×C20, C2×C20 [×2], C22×D5 [×2], C4×Dic3, D6⋊C4 [×2], C3×C4⋊C4, S3×C2×C4 [×2], C2×D12, Dic15 [×2], Dic15, C60 [×2], C60 [×2], D30 [×4], D30 [×4], C2×C30, C4×Dic5, D10⋊C4 [×2], C5×C4⋊C4, C2×C4×D5 [×2], C2×D20, Dic35D4, C4×D15 [×4], D60 [×4], C2×Dic15 [×2], C2×C60, C2×C60 [×2], C22×D15 [×2], D208C4, C4×Dic15, D303C4 [×2], C15×C4⋊C4, C2×C4×D15 [×2], C2×D60, D6011C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×2], C23, D5, D6 [×3], C22×C4, C2×D4, C4○D4, D10 [×3], C4×S3 [×2], C22×S3, D15, C4×D4, C4×D5 [×2], C22×D5, S3×C2×C4, S3×D4, Q83S3, D30 [×3], C2×C4×D5, D4×D5, Q82D5, Dic35D4, C4×D15 [×2], C22×D15, D208C4, C2×C4×D15, D4×D15, Q83D15, D6011C4

Smallest permutation representation of D6011C4
On 240 points
Generators in S240
(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)
(1 238)(2 237)(3 236)(4 235)(5 234)(6 233)(7 232)(8 231)(9 230)(10 229)(11 228)(12 227)(13 226)(14 225)(15 224)(16 223)(17 222)(18 221)(19 220)(20 219)(21 218)(22 217)(23 216)(24 215)(25 214)(26 213)(27 212)(28 211)(29 210)(30 209)(31 208)(32 207)(33 206)(34 205)(35 204)(36 203)(37 202)(38 201)(39 200)(40 199)(41 198)(42 197)(43 196)(44 195)(45 194)(46 193)(47 192)(48 191)(49 190)(50 189)(51 188)(52 187)(53 186)(54 185)(55 184)(56 183)(57 182)(58 181)(59 240)(60 239)(61 163)(62 162)(63 161)(64 160)(65 159)(66 158)(67 157)(68 156)(69 155)(70 154)(71 153)(72 152)(73 151)(74 150)(75 149)(76 148)(77 147)(78 146)(79 145)(80 144)(81 143)(82 142)(83 141)(84 140)(85 139)(86 138)(87 137)(88 136)(89 135)(90 134)(91 133)(92 132)(93 131)(94 130)(95 129)(96 128)(97 127)(98 126)(99 125)(100 124)(101 123)(102 122)(103 121)(104 180)(105 179)(106 178)(107 177)(108 176)(109 175)(110 174)(111 173)(112 172)(113 171)(114 170)(115 169)(116 168)(117 167)(118 166)(119 165)(120 164)
(1 76 239 149)(2 107 240 180)(3 78 181 151)(4 109 182 122)(5 80 183 153)(6 111 184 124)(7 82 185 155)(8 113 186 126)(9 84 187 157)(10 115 188 128)(11 86 189 159)(12 117 190 130)(13 88 191 161)(14 119 192 132)(15 90 193 163)(16 61 194 134)(17 92 195 165)(18 63 196 136)(19 94 197 167)(20 65 198 138)(21 96 199 169)(22 67 200 140)(23 98 201 171)(24 69 202 142)(25 100 203 173)(26 71 204 144)(27 102 205 175)(28 73 206 146)(29 104 207 177)(30 75 208 148)(31 106 209 179)(32 77 210 150)(33 108 211 121)(34 79 212 152)(35 110 213 123)(36 81 214 154)(37 112 215 125)(38 83 216 156)(39 114 217 127)(40 85 218 158)(41 116 219 129)(42 87 220 160)(43 118 221 131)(44 89 222 162)(45 120 223 133)(46 91 224 164)(47 62 225 135)(48 93 226 166)(49 64 227 137)(50 95 228 168)(51 66 229 139)(52 97 230 170)(53 68 231 141)(54 99 232 172)(55 70 233 143)(56 101 234 174)(57 72 235 145)(58 103 236 176)(59 74 237 147)(60 105 238 178)

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

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

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

90 conjugacy classes

class 1 2A2B2C2D2E2F2G 3 4A···4F4G4H4I4J4K4L5A5B6A6B6C10A···10F12A···12F15A15B15C15D20A···20L30A···30L60A···60X
order1222222234···44444445566610···1012···121515151520···2030···3060···60
size11113030303022···2151515153030222222···24···422224···42···24···4

90 irreducible representations

dim111111122222222222444444
type+++++++++++++++++++
imageC1C2C2C2C2C2C4S3D4D5D6C4○D4D10C4×S3D15C4×D5D30C4×D15S3×D4Q83S3D4×D5Q82D5D4×D15Q83D15
kernelD6011C4C4×Dic15D303C4C15×C4⋊C4C2×C4×D15C2×D60D60C5×C4⋊C4Dic15C3×C4⋊C4C2×C20C30C2×C12C20C4⋊C4C12C2×C4C4C10C10C6C6C2C2
# reps11212181223264481216112244

Matrix representation of D6011C4 in GL6(𝔽61)

31140000
47370000
001100
0060000
00005716
0000184
,
33370000
25280000
00606000
000100
00005716
0000414
,
6000000
0600000
0050000
0005000
0000600
0000301

G:=sub<GL(6,GF(61))| [31,47,0,0,0,0,14,37,0,0,0,0,0,0,1,60,0,0,0,0,1,0,0,0,0,0,0,0,57,18,0,0,0,0,16,4],[33,25,0,0,0,0,37,28,0,0,0,0,0,0,60,0,0,0,0,0,60,1,0,0,0,0,0,0,57,41,0,0,0,0,16,4],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,50,0,0,0,0,0,0,50,0,0,0,0,0,0,60,30,0,0,0,0,0,1] >;

D6011C4 in GAP, Magma, Sage, TeX

D_{60}\rtimes_{11}C_4
% in TeX

G:=Group("D60:11C4");
// GroupNames label

G:=SmallGroup(480,858);
// by ID

G=gap.SmallGroup(480,858);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,253,120,219,58,2693,18822]);
// Polycyclic

G:=Group<a,b,c|a^60=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^31,c*b*c^-1=a^30*b>;
// generators/relations

׿
×
𝔽