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G = C60.68D4order 480 = 25·3·5

68th non-split extension by C60 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C60.68D4, D107Dic6, C12.53D20, (C6×D5)⋊7Q8, C4⋊Dic34D5, C6.28(Q8×D5), C6.55(C2×D20), C154(C22⋊Q8), C30.22(C2×Q8), C35(D102Q8), C30.109(C2×D4), (C2×C20).112D6, C6.Dic104C2, C2.12(D5×Dic6), (C2×Dic30)⋊27C2, C30.27(C4○D4), (C2×C12).297D10, C4.18(C3⋊D20), C20.34(C3⋊D4), C52(C12.48D4), (C2×C30).50C23, C10.10(C2×Dic6), (C22×D5).84D6, C10.51(C4○D12), C6.20(D42D5), C2.8(D125D5), (C2×C60).141C22, (C2×Dic3).11D10, (C2×Dic5).161D6, D10⋊Dic3.4C2, (C10×Dic3).30C22, (C2×Dic15).52C22, (C6×Dic5).183C22, (C2×C4×D5).4S3, (D5×C2×C12).4C2, (C5×C4⋊Dic3)⋊3C2, (C2×C4).154(S3×D5), C2.14(C2×C3⋊D20), C10.10(C2×C3⋊D4), (D5×C2×C6).97C22, C22.137(C2×S3×D5), (C2×C6).62(C22×D5), (C2×C10).62(C22×S3), SmallGroup(480,436)

Series: Derived Chief Lower central Upper central

C1C2×C30 — C60.68D4
C1C5C15C30C2×C30D5×C2×C6D10⋊Dic3 — C60.68D4
C15C2×C30 — C60.68D4
C1C22C2×C4

Generators and relations for C60.68D4
 G = < a,b,c | a60=b4=1, c2=a30, bab-1=a11, cac-1=a-1, cbc-1=a30b-1 >

Subgroups: 700 in 148 conjugacy classes, 56 normal (34 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×2], C4 [×5], C22, C22 [×4], C5, C6 [×3], C6 [×2], C2×C4, C2×C4 [×7], Q8 [×2], C23, D5 [×2], C10 [×3], Dic3 [×4], C12 [×2], C12, C2×C6, C2×C6 [×4], C15, C22⋊C4 [×2], C4⋊C4 [×3], C22×C4, C2×Q8, Dic5 [×3], C20 [×2], C20 [×2], D10 [×2], D10 [×2], C2×C10, Dic6 [×2], C2×Dic3 [×2], C2×Dic3 [×2], C2×C12, C2×C12 [×3], C22×C6, C3×D5 [×2], C30 [×3], C22⋊Q8, Dic10 [×2], C4×D5 [×2], C2×Dic5, C2×Dic5 [×2], C2×C20, C2×C20 [×2], C22×D5, Dic3⋊C4 [×2], C4⋊Dic3, C6.D4 [×2], C2×Dic6, C22×C12, C5×Dic3 [×2], C3×Dic5, Dic15 [×2], C60 [×2], C6×D5 [×2], C6×D5 [×2], C2×C30, C4⋊Dic5 [×2], D10⋊C4 [×2], C5×C4⋊C4, C2×Dic10, C2×C4×D5, C12.48D4, D5×C12 [×2], C6×Dic5, C10×Dic3 [×2], Dic30 [×2], C2×Dic15 [×2], C2×C60, D5×C2×C6, D102Q8, D10⋊Dic3 [×2], C6.Dic10 [×2], C5×C4⋊Dic3, D5×C2×C12, C2×Dic30, C60.68D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], Q8 [×2], C23, D5, D6 [×3], C2×D4, C2×Q8, C4○D4, D10 [×3], Dic6 [×2], C3⋊D4 [×2], C22×S3, C22⋊Q8, D20 [×2], C22×D5, C2×Dic6, C4○D12, C2×C3⋊D4, S3×D5, C2×D20, D42D5, Q8×D5, C12.48D4, C3⋊D20 [×2], C2×S3×D5, D102Q8, D5×Dic6, D125D5, C2×C3⋊D20, C60.68D4

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

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

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

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

66 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D4E4F4G4H5A5B6A6B6C6D6E6F6G10A···10F12A12B12C12D12E12F12G12H15A15B20A20B20C20D20E···20L30A···30F60A···60H
order12222234444444455666666610···10121212121212121215152020202020···2030···3060···60
size1111101022210101212606022222101010102···222221010101044444412···124···44···4

66 irreducible representations

dim111111222222222222224444444
type++++++++-++++++-++--++--
imageC1C2C2C2C2C2S3D4Q8D5D6D6D6C4○D4D10D10C3⋊D4Dic6D20C4○D12S3×D5D42D5Q8×D5C3⋊D20C2×S3×D5D5×Dic6D125D5
kernelC60.68D4D10⋊Dic3C6.Dic10C5×C4⋊Dic3D5×C2×C12C2×Dic30C2×C4×D5C60C6×D5C4⋊Dic3C2×Dic5C2×C20C22×D5C30C2×Dic3C2×C12C20D10C12C10C2×C4C6C6C4C22C2C2
# reps122111122211124244842224244

Matrix representation of C60.68D4 in GL4(𝔽61) generated by

21000
443200
00117
004417
,
111800
345000
00297
005432
,
111800
05000
0022
002959
G:=sub<GL(4,GF(61))| [21,44,0,0,0,32,0,0,0,0,1,44,0,0,17,17],[11,34,0,0,18,50,0,0,0,0,29,54,0,0,7,32],[11,0,0,0,18,50,0,0,0,0,2,29,0,0,2,59] >;

C60.68D4 in GAP, Magma, Sage, TeX

C_{60}._{68}D_4
% in TeX

G:=Group("C60.68D4");
// GroupNames label

G:=SmallGroup(480,436);
// by ID

G=gap.SmallGroup(480,436);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,141,176,219,100,1356,18822]);
// Polycyclic

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

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