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G = C60.46D4order 480 = 25·3·5

46th non-split extension by C60 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C60.46D4, D67Dic10, (S3×C10)⋊7Q8, C605C424C2, C54(D63Q8), C156(C22⋊Q8), C30.24(C2×Q8), C10.28(S3×Q8), (C2×Dic10)⋊3S3, (C6×Dic10)⋊5C2, C6.6(C4○D20), C30.113(C2×D4), (C2×C20).294D6, D6⋊Dic5.7C2, Dic155C48C2, C30.35(C4○D4), (C2×C12).115D10, C20.83(C3⋊D4), C34(C20.48D4), C4.25(C15⋊D4), C12.37(C5⋊D4), (C2×C30).59C23, (C2×Dic5).16D6, C2.12(S3×Dic10), C6.10(C2×Dic10), (C2×C60).113C22, C10.8(Q83S3), (C22×S3).67D10, C2.11(D60⋊C2), (C2×Dic3).141D10, (C6×Dic5).35C22, (C2×Dic15).58C22, (C10×Dic3).165C22, (S3×C2×C4).4D5, (S3×C2×C20).4C2, C6.82(C2×C5⋊D4), (C2×C4).104(S3×D5), C2.16(C2×C15⋊D4), C10.83(C2×C3⋊D4), C22.146(C2×S3×D5), (S3×C2×C10).79C22, (C2×C6).71(C22×D5), (C2×C10).71(C22×S3), SmallGroup(480,445)

Series: Derived Chief Lower central Upper central

C1C2×C30 — C60.46D4
C1C5C15C30C2×C30C6×Dic5D6⋊Dic5 — C60.46D4
C15C2×C30 — C60.46D4
C1C22C2×C4

Generators and relations for C60.46D4
 G = < a,b,c | a60=b4=1, c2=a30, bab-1=a-1, cac-1=a19, cbc-1=a30b-1 >

Subgroups: 604 in 148 conjugacy classes, 56 normal (34 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×2], C4 [×5], C22, C22 [×4], C5, S3 [×2], C6 [×3], C2×C4, C2×C4 [×7], Q8 [×2], C23, C10 [×3], C10 [×2], Dic3 [×3], C12 [×2], C12 [×2], D6 [×2], D6 [×2], C2×C6, C15, C22⋊C4 [×2], C4⋊C4 [×3], C22×C4, C2×Q8, Dic5 [×4], C20 [×2], C20, C2×C10, C2×C10 [×4], C4×S3 [×2], C2×Dic3, C2×Dic3 [×2], C2×C12, C2×C12 [×2], C3×Q8 [×2], C22×S3, C5×S3 [×2], C30 [×3], C22⋊Q8, Dic10 [×2], C2×Dic5 [×2], C2×Dic5 [×2], C2×C20, C2×C20 [×3], C22×C10, Dic3⋊C4 [×2], C4⋊Dic3, D6⋊C4 [×2], S3×C2×C4, C6×Q8, C5×Dic3, C3×Dic5 [×2], Dic15 [×2], C60 [×2], S3×C10 [×2], S3×C10 [×2], C2×C30, C10.D4 [×2], C4⋊Dic5, C23.D5 [×2], C2×Dic10, C22×C20, D63Q8, C3×Dic10 [×2], C6×Dic5 [×2], S3×C20 [×2], C10×Dic3, C2×Dic15 [×2], C2×C60, S3×C2×C10, C20.48D4, D6⋊Dic5 [×2], Dic155C4 [×2], C605C4, C6×Dic10, S3×C2×C20, C60.46D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], Q8 [×2], C23, D5, D6 [×3], C2×D4, C2×Q8, C4○D4, D10 [×3], C3⋊D4 [×2], C22×S3, C22⋊Q8, Dic10 [×2], C5⋊D4 [×2], C22×D5, S3×Q8, Q83S3, C2×C3⋊D4, S3×D5, C2×Dic10, C4○D20, C2×C5⋊D4, D63Q8, C15⋊D4 [×2], C2×S3×D5, C20.48D4, S3×Dic10, D60⋊C2, C2×C15⋊D4, C60.46D4

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

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

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

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

72 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D4E4F4G4H5A5B6A6B6C10A···10F10G···10N12A12B12C12D12E12F15A15B20A···20H20I···20P30A···30F60A···60H
order1222223444444445566610···1010···10121212121212151520···2020···2030···3060···60
size1111662226620206060222222···26···64420202020442···26···64···44···4

72 irreducible representations

dim111111222222222222224444444
type++++++++-++++++--++-+-+
imageC1C2C2C2C2C2S3D4Q8D5D6D6C4○D4D10D10D10C3⋊D4C5⋊D4Dic10C4○D20S3×Q8Q83S3S3×D5C15⋊D4C2×S3×D5S3×Dic10D60⋊C2
kernelC60.46D4D6⋊Dic5Dic155C4C605C4C6×Dic10S3×C2×C20C2×Dic10C60S3×C10S3×C2×C4C2×Dic5C2×C20C30C2×Dic3C2×C12C22×S3C20C12D6C6C10C10C2×C4C4C22C2C2
# reps122111122221222248881124244

Matrix representation of C60.46D4 in GL4(𝔽61) generated by

0100
606000
00230
00228
,
524300
52900
002428
006037
,
524300
18900
002428
006037
G:=sub<GL(4,GF(61))| [0,60,0,0,1,60,0,0,0,0,23,22,0,0,0,8],[52,52,0,0,43,9,0,0,0,0,24,60,0,0,28,37],[52,18,0,0,43,9,0,0,0,0,24,60,0,0,28,37] >;

C60.46D4 in GAP, Magma, Sage, TeX

C_{60}._{46}D_4
% in TeX

G:=Group("C60.46D4");
// GroupNames label

G:=SmallGroup(480,445);
// by ID

G=gap.SmallGroup(480,445);
# by ID

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

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

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