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G = C60.63D4order 480 = 25·3·5

63rd non-split extension by C60 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C60.63D4, D20.33D6, C12.52D20, C60.106C23, Dic10.33D6, Dic30.47C22, C3⋊C8.3D10, (C2×C6).7D20, C4○D20.5S3, (C2×C20).97D6, C6.51(C2×D20), (C2×C30).56D4, C30.88(C2×D4), C4.Dic39D5, C3⋊Dic2013C2, (C2×C12).99D10, C51(Q8.14D6), C35(C8.D10), C159(C8.C22), (C2×Dic30)⋊18C2, C6.D2014C2, C20.30(C3⋊D4), C4.17(C3⋊D20), (C2×C60).95C22, C12.97(C22×D5), C20.156(C22×S3), (C3×D20).38C22, C22.10(C3⋊D20), (C3×Dic10).38C22, C4.105(C2×S3×D5), (C2×C4).15(S3×D5), C10.6(C2×C3⋊D4), (C3×C4○D20).4C2, C2.10(C2×C3⋊D20), (C5×C3⋊C8).21C22, (C5×C4.Dic3)⋊9C2, (C2×C10).13(C3⋊D4), SmallGroup(480,389)

Series: Derived Chief Lower central Upper central

C1C60 — C60.63D4
C1C5C15C30C60C3×D20C6.D20 — C60.63D4
C15C30C60 — C60.63D4
C1C2C2×C4

Generators and relations for C60.63D4
 G = < a,b,c | a60=1, b4=c2=a30, bab-1=a11, cac-1=a-1, cbc-1=b3 >

Subgroups: 620 in 120 conjugacy classes, 44 normal (34 characteristic)
C1, C2, C2 [×2], C3, C4 [×2], C4 [×3], C22, C22, C5, C6, C6 [×2], C8 [×2], C2×C4, C2×C4 [×2], D4 [×2], Q8 [×4], D5, C10, C10, Dic3 [×2], C12 [×2], C12, C2×C6, C2×C6, C15, M4(2), SD16 [×2], Q16 [×2], C2×Q8, C4○D4, Dic5 [×3], C20 [×2], D10, C2×C10, C3⋊C8 [×2], Dic6 [×3], C2×Dic3, C2×C12, C2×C12, C3×D4 [×2], C3×Q8, C3×D5, C30, C30, C8.C22, C40 [×2], Dic10, Dic10 [×3], C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C4.Dic3, D4.S3 [×2], C3⋊Q16 [×2], C2×Dic6, C3×C4○D4, C3×Dic5, Dic15 [×2], C60 [×2], C6×D5, C2×C30, C40⋊C2 [×2], Dic20 [×2], C5×M4(2), C2×Dic10, C4○D20, Q8.14D6, C5×C3⋊C8 [×2], C3×Dic10, D5×C12, C3×D20, C3×C5⋊D4, Dic30 [×2], Dic30, C2×Dic15, C2×C60, C8.D10, C6.D20 [×2], C3⋊Dic20 [×2], C5×C4.Dic3, C3×C4○D20, C2×Dic30, C60.63D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, D10 [×3], C3⋊D4 [×2], C22×S3, C8.C22, D20 [×2], C22×D5, C2×C3⋊D4, S3×D5, C2×D20, Q8.14D6, C3⋊D20 [×2], C2×S3×D5, C8.D10, C2×C3⋊D20, C60.63D4

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

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

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

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

57 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D4E5A5B6A6B6C6D8A8B10A10B10C10D12A12B12C12D12E15A15B20A20B20C20D20E20F30A···30F40A···40H60A···60H
order122234444455666688101010101212121212151520202020202030···3040···4060···60
size1122022220606022242020121222442242020442222444···412···124···4

57 irreducible representations

dim111111222222222222244444444
type+++++++++++++++++-+-+++--
imageC1C2C2C2C2C2S3D4D4D5D6D6D6D10D10C3⋊D4C3⋊D4D20D20C8.C22S3×D5Q8.14D6C3⋊D20C2×S3×D5C3⋊D20C8.D10C60.63D4
kernelC60.63D4C6.D20C3⋊Dic20C5×C4.Dic3C3×C4○D20C2×Dic30C4○D20C60C2×C30C4.Dic3Dic10D20C2×C20C3⋊C8C2×C12C20C2×C10C12C2×C6C15C2×C4C5C4C4C22C3C1
# reps122111111211142224412222248

Matrix representation of C60.63D4 in GL4(𝔽241) generated by

7822000
14917000
21022057
657184235
,
002401
190123951
831782400
1241812400
,
145900
1322700
22321233102
227498208
G:=sub<GL(4,GF(241))| [78,149,21,6,220,170,0,57,0,0,220,184,0,0,57,235],[0,190,83,124,0,1,178,181,240,239,240,240,1,51,0,0],[14,13,223,227,59,227,212,4,0,0,33,98,0,0,102,208] >;

C60.63D4 in GAP, Magma, Sage, TeX

C_{60}._{63}D_4
% in TeX

G:=Group("C60.63D4");
// GroupNames label

G:=SmallGroup(480,389);
// by ID

G=gap.SmallGroup(480,389);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,141,64,219,100,675,80,1356,18822]);
// Polycyclic

G:=Group<a,b,c|a^60=1,b^4=c^2=a^30,b*a*b^-1=a^11,c*a*c^-1=a^-1,c*b*c^-1=b^3>;
// generators/relations

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