Copied to
clipboard

G = C60.89D4order 480 = 25·3·5

89th non-split extension by C60 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C60.89D4, D6⋊Dic55C2, (C2×D12).8D5, (C10×D12).8C2, (C2×C20).114D6, C30.114(C2×D4), C156(C4.4D4), (C2×Dic10)⋊10S3, (C6×Dic10)⋊10C2, (C4×Dic15)⋊24C2, C30.36(C4○D4), (C2×C12).116D10, C6.8(D42D5), C53(C12.23D4), C20.36(C3⋊D4), C33(C20.17D4), C4.10(C15⋊D4), C12.38(C5⋊D4), (C2×C30).60C23, (C2×Dic5).17D6, (C22×S3).7D10, (C2×C60).195C22, C2.12(D12⋊D5), C10.27(Q83S3), (C6×Dic5).36C22, (C2×Dic15).189C22, C6.83(C2×C5⋊D4), (C2×C4).205(S3×D5), C10.84(C2×C3⋊D4), C2.17(C2×C15⋊D4), (S3×C2×C10).7C22, C22.147(C2×S3×D5), (C2×C6).72(C22×D5), (C2×C10).72(C22×S3), SmallGroup(480,446)

Series: Derived Chief Lower central Upper central

C1C2×C30 — C60.89D4
C1C5C15C30C2×C30C6×Dic5D6⋊Dic5 — C60.89D4
C15C2×C30 — C60.89D4
C1C22C2×C4

Generators and relations for C60.89D4
 G = < a,b,c | a60=b4=1, c2=a30, bab-1=a29, cac-1=a19, cbc-1=a30b-1 >

Subgroups: 668 in 152 conjugacy classes, 52 normal (22 characteristic)
C1, C2, C2 [×2], C2 [×2], C3, C4 [×2], C4 [×4], C22, C22 [×6], C5, S3 [×2], C6, C6 [×2], C2×C4, C2×C4 [×4], D4 [×2], Q8 [×2], C23 [×2], C10, C10 [×2], C10 [×2], Dic3 [×2], C12 [×2], C12 [×2], D6 [×6], C2×C6, C15, C42, C22⋊C4 [×4], C2×D4, C2×Q8, Dic5 [×4], C20 [×2], C2×C10, C2×C10 [×6], D12 [×2], C2×Dic3 [×2], C2×C12, C2×C12 [×2], C3×Q8 [×2], C22×S3 [×2], C5×S3 [×2], C30, C30 [×2], C4.4D4, Dic10 [×2], C2×Dic5 [×2], C2×Dic5 [×2], C2×C20, C5×D4 [×2], C22×C10 [×2], C4×Dic3, D6⋊C4 [×4], C2×D12, C6×Q8, C3×Dic5 [×2], Dic15 [×2], C60 [×2], S3×C10 [×6], C2×C30, C4×Dic5, C23.D5 [×4], C2×Dic10, D4×C10, C12.23D4, C3×Dic10 [×2], C6×Dic5 [×2], C5×D12 [×2], C2×Dic15 [×2], C2×C60, S3×C2×C10 [×2], C20.17D4, D6⋊Dic5 [×4], C4×Dic15, C6×Dic10, C10×D12, C60.89D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, C4○D4 [×2], D10 [×3], C3⋊D4 [×2], C22×S3, C4.4D4, C5⋊D4 [×2], C22×D5, Q83S3 [×2], C2×C3⋊D4, S3×D5, D42D5 [×2], C2×C5⋊D4, C12.23D4, C15⋊D4 [×2], C2×S3×D5, C20.17D4, D12⋊D5 [×2], C2×C15⋊D4, C60.89D4

Smallest permutation representation of C60.89D4
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)
(1 184 71 125)(2 213 72 154)(3 182 73 123)(4 211 74 152)(5 240 75 121)(6 209 76 150)(7 238 77 179)(8 207 78 148)(9 236 79 177)(10 205 80 146)(11 234 81 175)(12 203 82 144)(13 232 83 173)(14 201 84 142)(15 230 85 171)(16 199 86 140)(17 228 87 169)(18 197 88 138)(19 226 89 167)(20 195 90 136)(21 224 91 165)(22 193 92 134)(23 222 93 163)(24 191 94 132)(25 220 95 161)(26 189 96 130)(27 218 97 159)(28 187 98 128)(29 216 99 157)(30 185 100 126)(31 214 101 155)(32 183 102 124)(33 212 103 153)(34 181 104 122)(35 210 105 151)(36 239 106 180)(37 208 107 149)(38 237 108 178)(39 206 109 147)(40 235 110 176)(41 204 111 145)(42 233 112 174)(43 202 113 143)(44 231 114 172)(45 200 115 141)(46 229 116 170)(47 198 117 139)(48 227 118 168)(49 196 119 137)(50 225 120 166)(51 194 61 135)(52 223 62 164)(53 192 63 133)(54 221 64 162)(55 190 65 131)(56 219 66 160)(57 188 67 129)(58 217 68 158)(59 186 69 127)(60 215 70 156)
(1 199 31 229)(2 218 32 188)(3 237 33 207)(4 196 34 226)(5 215 35 185)(6 234 36 204)(7 193 37 223)(8 212 38 182)(9 231 39 201)(10 190 40 220)(11 209 41 239)(12 228 42 198)(13 187 43 217)(14 206 44 236)(15 225 45 195)(16 184 46 214)(17 203 47 233)(18 222 48 192)(19 181 49 211)(20 200 50 230)(21 219 51 189)(22 238 52 208)(23 197 53 227)(24 216 54 186)(25 235 55 205)(26 194 56 224)(27 213 57 183)(28 232 58 202)(29 191 59 221)(30 210 60 240)(61 130 91 160)(62 149 92 179)(63 168 93 138)(64 127 94 157)(65 146 95 176)(66 165 96 135)(67 124 97 154)(68 143 98 173)(69 162 99 132)(70 121 100 151)(71 140 101 170)(72 159 102 129)(73 178 103 148)(74 137 104 167)(75 156 105 126)(76 175 106 145)(77 134 107 164)(78 153 108 123)(79 172 109 142)(80 131 110 161)(81 150 111 180)(82 169 112 139)(83 128 113 158)(84 147 114 177)(85 166 115 136)(86 125 116 155)(87 144 117 174)(88 163 118 133)(89 122 119 152)(90 141 120 171)

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

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

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

60 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D4E4F4G4H5A5B6A6B6C10A···10F10G···10N12A12B12C12D12E12F15A15B20A20B20C20D30A···30F60A···60H
order1222223444444445566610···1010···1012121212121215152020202030···3060···60
size11111212222202030303030222222···212···1244202020204444444···44···4

60 irreducible representations

dim111112222222222444444
type++++++++++++++--+
imageC1C2C2C2C2S3D4D5D6D6C4○D4D10D10C3⋊D4C5⋊D4Q83S3S3×D5D42D5C15⋊D4C2×S3×D5D12⋊D5
kernelC60.89D4D6⋊Dic5C4×Dic15C6×Dic10C10×D12C2×Dic10C60C2×D12C2×Dic5C2×C20C30C2×C12C22×S3C20C12C10C2×C4C6C4C22C2
# reps141111222142448224428

Matrix representation of C60.89D4 in GL6(𝔽61)

300000
26410000
0016000
001000
0000608
0000151
,
1120000
0500000
000100
001000
00001134
00001850
,
1120000
1500000
0060000
0006000
0000500
00004311

G:=sub<GL(6,GF(61))| [3,26,0,0,0,0,0,41,0,0,0,0,0,0,1,1,0,0,0,0,60,0,0,0,0,0,0,0,60,15,0,0,0,0,8,1],[11,0,0,0,0,0,2,50,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,11,18,0,0,0,0,34,50],[11,1,0,0,0,0,2,50,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,50,43,0,0,0,0,0,11] >;

C60.89D4 in GAP, Magma, Sage, TeX

C_{60}._{89}D_4
% in TeX

G:=Group("C60.89D4");
// GroupNames label

G:=SmallGroup(480,446);
// by ID

G=gap.SmallGroup(480,446);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,141,422,219,100,1356,18822]);
// Polycyclic

G:=Group<a,b,c|a^60=b^4=1,c^2=a^30,b*a*b^-1=a^29,c*a*c^-1=a^19,c*b*c^-1=a^30*b^-1>;
// generators/relations

׿
×
𝔽