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G = C60.70D4order 480 = 25·3·5

70th non-split extension by C60 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C60.70D4, C20.20D12, (C4×Dic5)⋊4S3, (C2×Dic6)⋊3D5, D304C49C2, C53(C427S3), (C10×Dic6)⋊6C2, (C12×Dic5)⋊4C2, (C2×D60).18C2, C30.117(C2×D4), (C2×C20).116D6, C10.58(C2×D12), C159(C4.4D4), C10.7(C4○D12), C30.41(C4○D4), (C2×C12).299D10, C4.10(C5⋊D12), C12.60(C5⋊D4), C31(C20.23D4), (C2×C30).65C23, C6.9(Q82D5), (C2×C60).143C22, (C2×Dic3).18D10, (C2×Dic5).165D6, C2.12(C12.28D10), (C10×Dic3).37C22, (C6×Dic5).187C22, (C22×D15).24C22, C6.12(C2×C5⋊D4), (C2×C4).156(S3×D5), C2.16(C2×C5⋊D12), C22.151(C2×S3×D5), (C2×C6).77(C22×D5), (C2×C10).77(C22×S3), SmallGroup(480,451)

Series: Derived Chief Lower central Upper central

C1C2×C30 — C60.70D4
C1C5C15C30C2×C30C6×Dic5D304C4 — C60.70D4
C15C2×C30 — C60.70D4
C1C22C2×C4

Generators and relations for C60.70D4
 G = < a,b,c | a60=b4=c2=1, bab-1=a49, cac=a-1, cbc=a30b-1 >

Subgroups: 972 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], D5 [×2], C10, C10 [×2], Dic3 [×2], C12 [×2], C12 [×2], D6 [×6], C2×C6, C15, C42, C22⋊C4 [×4], C2×D4, C2×Q8, Dic5 [×2], C20 [×2], C20 [×2], D10 [×6], C2×C10, Dic6 [×2], D12 [×2], C2×Dic3 [×2], C2×C12, C2×C12 [×2], C22×S3 [×2], D15 [×2], C30, C30 [×2], C4.4D4, D20 [×2], C2×Dic5 [×2], C2×C20, C2×C20 [×2], C5×Q8 [×2], C22×D5 [×2], D6⋊C4 [×4], C4×C12, C2×Dic6, C2×D12, C5×Dic3 [×2], C3×Dic5 [×2], C60 [×2], D30 [×6], C2×C30, C4×Dic5, D10⋊C4 [×4], C2×D20, Q8×C10, C427S3, C6×Dic5 [×2], C5×Dic6 [×2], C10×Dic3 [×2], D60 [×2], C2×C60, C22×D15 [×2], C20.23D4, D304C4 [×4], C12×Dic5, C10×Dic6, C2×D60, C60.70D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, C4○D4 [×2], D10 [×3], D12 [×2], C22×S3, C4.4D4, C5⋊D4 [×2], C22×D5, C2×D12, C4○D12 [×2], S3×D5, Q82D5 [×2], C2×C5⋊D4, C427S3, C5⋊D12 [×2], C2×S3×D5, C20.23D4, C12.28D10 [×2], C2×C5⋊D12, C60.70D4

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

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

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

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

66 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D4E4F4G4H5A5B6A6B6C10A···10F12A12B12C12D12E···12L15A15B20A20B20C20D20E···20L30A···30F60A···60H
order1222223444444445566610···101212121212···1215152020202020···2030···3060···60
size11116060222101010101212222222···2222210···1044444412···124···44···4

66 irreducible representations

dim111112222222222244444
type++++++++++++++++++
imageC1C2C2C2C2S3D4D5D6D6C4○D4D10D10D12C5⋊D4C4○D12S3×D5Q82D5C5⋊D12C2×S3×D5C12.28D10
kernelC60.70D4D304C4C12×Dic5C10×Dic6C2×D60C4×Dic5C60C2×Dic6C2×Dic5C2×C20C30C2×Dic3C2×C12C20C12C10C2×C4C6C4C22C2
# reps141111222144248824428

Matrix representation of C60.70D4 in GL6(𝔽61)

010000
60600000
0011700
00441700
00001538
00002338
,
23460000
15380000
00174400
00604400
0000918
00004352
,
6000000
110000
00441700
0011700
00002346
00002338

G:=sub<GL(6,GF(61))| [0,60,0,0,0,0,1,60,0,0,0,0,0,0,1,44,0,0,0,0,17,17,0,0,0,0,0,0,15,23,0,0,0,0,38,38],[23,15,0,0,0,0,46,38,0,0,0,0,0,0,17,60,0,0,0,0,44,44,0,0,0,0,0,0,9,43,0,0,0,0,18,52],[60,1,0,0,0,0,0,1,0,0,0,0,0,0,44,1,0,0,0,0,17,17,0,0,0,0,0,0,23,23,0,0,0,0,46,38] >;

C60.70D4 in GAP, Magma, Sage, TeX

C_{60}._{70}D_4
% in TeX

G:=Group("C60.70D4");
// GroupNames label

G:=SmallGroup(480,451);
// by ID

G=gap.SmallGroup(480,451);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,365,120,219,100,1356,18822]);
// Polycyclic

G:=Group<a,b,c|a^60=b^4=c^2=1,b*a*b^-1=a^49,c*a*c=a^-1,c*b*c=a^30*b^-1>;
// generators/relations

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