Copied to
clipboard

G = C605C4⋊C2order 480 = 25·3·5

2nd semidirect product of C605C4 and C2 acting faithfully

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D6⋊C4.2D5, C605C42C2, (C2×C20).7D6, (C2×C12).7D10, C6.4(C4○D20), C10.D44S3, D6⋊Dic5.3C2, (C2×C60).3C22, Dic155C41C2, (C2×Dic5).5D6, C153(C422C2), (Dic3×Dic5)⋊5C2, C30.11(C4○D4), (C2×C30).32C23, (C22×S3).3D10, C10.49(C4○D12), C6.18(D42D5), C2.7(D125D5), C2.8(D60⋊C2), C10.5(Q83S3), (C2×Dic3).82D10, C32(C23.D10), C10.63(D42S3), C2.9(C30.C23), (C6×Dic5).17C22, (C2×Dic15).39C22, (C10×Dic3).16C22, C55(C4⋊C4⋊S3), (C5×D6⋊C4).2C2, (C2×C4).29(S3×D5), (S3×C2×C10).3C22, C22.123(C2×S3×D5), (C3×C10.D4)⋊4C2, (C2×C6).44(C22×D5), (C2×C10).44(C22×S3), SmallGroup(480,418)

Series: Derived Chief Lower central Upper central

C1C2×C30 — C605C4⋊C2
C1C5C15C30C2×C30C6×Dic5Dic3×Dic5 — C605C4⋊C2
C15C2×C30 — C605C4⋊C2
C1C22C2×C4

Generators and relations for C605C4⋊C2
 G = < a,b,c | a60=b4=c2=1, bab-1=a-1, cac=a41b2, cbc=a30b >

Subgroups: 524 in 120 conjugacy classes, 44 normal (all characteristic)
C1, C2 [×3], C2, C3, C4 [×6], C22, C22 [×3], C5, S3, C6 [×3], C2×C4, C2×C4 [×5], C23, C10 [×3], C10, Dic3 [×3], C12 [×3], D6 [×3], C2×C6, C15, C42, C22⋊C4 [×3], C4⋊C4 [×3], Dic5 [×4], C20 [×2], C2×C10, C2×C10 [×3], C2×Dic3, C2×Dic3 [×2], C2×C12, C2×C12 [×2], C22×S3, C5×S3, C30 [×3], C422C2, C2×Dic5 [×2], C2×Dic5 [×2], C2×C20, C2×C20, C22×C10, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, D6⋊C4 [×2], C3×C4⋊C4, C5×Dic3, C3×Dic5 [×2], Dic15 [×2], C60, S3×C10 [×3], C2×C30, C4×Dic5, C10.D4, C10.D4, C4⋊Dic5, C23.D5 [×2], C5×C22⋊C4, C4⋊C4⋊S3, C6×Dic5 [×2], C10×Dic3, C2×Dic15 [×2], C2×C60, S3×C2×C10, C23.D10, Dic3×Dic5, D6⋊Dic5 [×2], Dic155C4, C3×C10.D4, C5×D6⋊C4, C605C4, C605C4⋊C2
Quotients: C1, C2 [×7], C22 [×7], S3, C23, D5, D6 [×3], C4○D4 [×3], D10 [×3], C22×S3, C422C2, C22×D5, C4○D12, D42S3, Q83S3, S3×D5, C4○D20, D42D5 [×2], C4⋊C4⋊S3, C2×S3×D5, C23.D10, D60⋊C2, D125D5, C30.C23, C605C4⋊C2

Smallest permutation representation of C605C4⋊C2
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)
(1 153 78 197)(2 152 79 196)(3 151 80 195)(4 150 81 194)(5 149 82 193)(6 148 83 192)(7 147 84 191)(8 146 85 190)(9 145 86 189)(10 144 87 188)(11 143 88 187)(12 142 89 186)(13 141 90 185)(14 140 91 184)(15 139 92 183)(16 138 93 182)(17 137 94 181)(18 136 95 240)(19 135 96 239)(20 134 97 238)(21 133 98 237)(22 132 99 236)(23 131 100 235)(24 130 101 234)(25 129 102 233)(26 128 103 232)(27 127 104 231)(28 126 105 230)(29 125 106 229)(30 124 107 228)(31 123 108 227)(32 122 109 226)(33 121 110 225)(34 180 111 224)(35 179 112 223)(36 178 113 222)(37 177 114 221)(38 176 115 220)(39 175 116 219)(40 174 117 218)(41 173 118 217)(42 172 119 216)(43 171 120 215)(44 170 61 214)(45 169 62 213)(46 168 63 212)(47 167 64 211)(48 166 65 210)(49 165 66 209)(50 164 67 208)(51 163 68 207)(52 162 69 206)(53 161 70 205)(54 160 71 204)(55 159 72 203)(56 158 73 202)(57 157 74 201)(58 156 75 200)(59 155 76 199)(60 154 77 198)
(2 119)(3 23)(4 81)(5 45)(6 103)(8 65)(9 29)(10 87)(11 51)(12 109)(14 71)(15 35)(16 93)(17 57)(18 115)(20 77)(21 41)(22 99)(24 61)(26 83)(27 47)(28 105)(30 67)(32 89)(33 53)(34 111)(36 73)(38 95)(39 59)(40 117)(42 79)(44 101)(46 63)(48 85)(50 107)(52 69)(54 91)(56 113)(58 75)(60 97)(62 82)(64 104)(68 88)(70 110)(74 94)(76 116)(80 100)(86 106)(92 112)(98 118)(121 131)(122 216)(123 153)(124 238)(125 175)(126 200)(127 137)(128 222)(129 159)(130 184)(132 206)(133 143)(134 228)(135 165)(136 190)(138 212)(139 149)(140 234)(141 171)(142 196)(144 218)(145 155)(146 240)(147 177)(148 202)(150 224)(151 161)(152 186)(154 208)(156 230)(157 167)(158 192)(160 214)(162 236)(163 173)(164 198)(166 220)(168 182)(169 179)(170 204)(172 226)(174 188)(176 210)(178 232)(180 194)(181 231)(183 193)(185 215)(187 237)(189 199)(191 221)(195 205)(197 227)(201 211)(203 233)(207 217)(209 239)(213 223)(219 229)(225 235)

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

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

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

60 conjugacy classes

class 1 2A2B2C2D 3 4A4B4C4D4E4F4G4H4I5A5B6A6B6C10A···10F10G10H10I10J12A12B12C12D12E12F15A15B20A20B20C20D20E20F20G20H30A···30F60A···60H
order1222234444444445566610···10101010101212121212121515202020202020202030···3060···60
size1111122466101020303060222222···2121212124420202020444444121212124···44···4

60 irreducible representations

dim1111111222222222244444444
type++++++++++++++-++-++--
imageC1C2C2C2C2C2C2S3D5D6D6C4○D4D10D10D10C4○D12C4○D20D42S3Q83S3S3×D5D42D5C2×S3×D5D60⋊C2D125D5C30.C23
kernelC605C4⋊C2Dic3×Dic5D6⋊Dic5Dic155C4C3×C10.D4C5×D6⋊C4C605C4C10.D4D6⋊C4C2×Dic5C2×C20C30C2×Dic3C2×C12C22×S3C10C6C10C10C2×C4C6C22C2C2C2
# reps1121111122162224811242444

Matrix representation of C605C4⋊C2 in GL4(𝔽61) generated by

8000
522300
002338
002346
,
35600
265800
004352
00918
,
1000
506000
00060
00600
G:=sub<GL(4,GF(61))| [8,52,0,0,0,23,0,0,0,0,23,23,0,0,38,46],[3,26,0,0,56,58,0,0,0,0,43,9,0,0,52,18],[1,50,0,0,0,60,0,0,0,0,0,60,0,0,60,0] >;

C605C4⋊C2 in GAP, Magma, Sage, TeX

C_{60}\rtimes_5C_4\rtimes C_2
% in TeX

G:=Group("C60:5C4:C2");
// GroupNames label

G:=SmallGroup(480,418);
// by ID

G=gap.SmallGroup(480,418);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,176,590,219,58,1356,18822]);
// Polycyclic

G:=Group<a,b,c|a^60=b^4=c^2=1,b*a*b^-1=a^-1,c*a*c=a^41*b^2,c*b*c=a^30*b>;
// generators/relations

׿
×
𝔽