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G = D6015C4order 480 = 25·3·5

9th semidirect product of D60 and C4 acting via C4/C2=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D6015C4, C60.56D4, C30.18D8, C10.11D24, C12.49D20, C30.6SD16, C4⋊Dic31D5, C20.35(C4×S3), C6.6(D4⋊D5), (C2×C20).49D6, (C2×C30).16D4, C12.31(C4×D5), C52(C2.D24), C6.1(Q8⋊D5), C158(D4⋊C4), C31(D206C4), C60.106(C2×C4), (C2×D60).16C2, (C2×C10).30D12, C2.2(C5⋊D24), C10.4(C24⋊C2), C10.20(D6⋊C4), (C2×C12).284D10, C20.11(C3⋊D4), C4.14(C3⋊D20), C4.6(D30.C2), C6.5(D10⋊C4), C2.6(D304C4), C30.52(C22⋊C4), (C2×C60).128C22, C2.1(Dic6⋊D5), C22.13(C5⋊D12), (C6×C52C8)⋊5C2, (C2×C52C8)⋊2S3, (C5×C4⋊Dic3)⋊1C2, (C2×C4).134(S3×D5), (C2×C6).25(C5⋊D4), SmallGroup(480,45)

Series: Derived Chief Lower central Upper central

C1C60 — D6015C4
C1C5C15C30C60C2×C60C6×C52C8 — D6015C4
C15C30C60 — D6015C4
C1C22C2×C4

Generators and relations for D6015C4
 G = < a,b,c | a60=b2=c4=1, bab=a-1, cac-1=a11, cbc-1=a25b >

Subgroups: 796 in 100 conjugacy classes, 40 normal (38 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×2], C4, C22, C22 [×4], C5, S3 [×2], C6 [×3], C8, C2×C4, C2×C4, D4 [×3], C23, D5 [×2], C10 [×3], Dic3, C12 [×2], D6 [×4], C2×C6, C15, C4⋊C4, C2×C8, C2×D4, C20 [×2], C20, D10 [×4], C2×C10, C24, D12 [×3], C2×Dic3, C2×C12, C22×S3, D15 [×2], C30 [×3], D4⋊C4, C52C8, D20 [×3], C2×C20, C2×C20, C22×D5, C4⋊Dic3, C2×C24, C2×D12, C5×Dic3, C60 [×2], D30 [×4], C2×C30, C2×C52C8, C5×C4⋊C4, C2×D20, C2.D24, C3×C52C8, C10×Dic3, D60 [×2], D60, C2×C60, C22×D15, D206C4, C6×C52C8, C5×C4⋊Dic3, C2×D60, D6015C4
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], D5, D6, C22⋊C4, D8, SD16, D10, C4×S3, D12, C3⋊D4, D4⋊C4, C4×D5, D20, C5⋊D4, C24⋊C2, D24, D6⋊C4, S3×D5, D10⋊C4, D4⋊D5, Q8⋊D5, C2.D24, D30.C2, C3⋊D20, C5⋊D12, D206C4, C5⋊D24, Dic6⋊D5, D304C4, D6015C4

Smallest permutation representation of D6015C4
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)
(1 127)(2 126)(3 125)(4 124)(5 123)(6 122)(7 121)(8 180)(9 179)(10 178)(11 177)(12 176)(13 175)(14 174)(15 173)(16 172)(17 171)(18 170)(19 169)(20 168)(21 167)(22 166)(23 165)(24 164)(25 163)(26 162)(27 161)(28 160)(29 159)(30 158)(31 157)(32 156)(33 155)(34 154)(35 153)(36 152)(37 151)(38 150)(39 149)(40 148)(41 147)(42 146)(43 145)(44 144)(45 143)(46 142)(47 141)(48 140)(49 139)(50 138)(51 137)(52 136)(53 135)(54 134)(55 133)(56 132)(57 131)(58 130)(59 129)(60 128)(61 238)(62 237)(63 236)(64 235)(65 234)(66 233)(67 232)(68 231)(69 230)(70 229)(71 228)(72 227)(73 226)(74 225)(75 224)(76 223)(77 222)(78 221)(79 220)(80 219)(81 218)(82 217)(83 216)(84 215)(85 214)(86 213)(87 212)(88 211)(89 210)(90 209)(91 208)(92 207)(93 206)(94 205)(95 204)(96 203)(97 202)(98 201)(99 200)(100 199)(101 198)(102 197)(103 196)(104 195)(105 194)(106 193)(107 192)(108 191)(109 190)(110 189)(111 188)(112 187)(113 186)(114 185)(115 184)(116 183)(117 182)(118 181)(119 240)(120 239)
(1 224 128 61)(2 235 129 72)(3 186 130 83)(4 197 131 94)(5 208 132 105)(6 219 133 116)(7 230 134 67)(8 181 135 78)(9 192 136 89)(10 203 137 100)(11 214 138 111)(12 225 139 62)(13 236 140 73)(14 187 141 84)(15 198 142 95)(16 209 143 106)(17 220 144 117)(18 231 145 68)(19 182 146 79)(20 193 147 90)(21 204 148 101)(22 215 149 112)(23 226 150 63)(24 237 151 74)(25 188 152 85)(26 199 153 96)(27 210 154 107)(28 221 155 118)(29 232 156 69)(30 183 157 80)(31 194 158 91)(32 205 159 102)(33 216 160 113)(34 227 161 64)(35 238 162 75)(36 189 163 86)(37 200 164 97)(38 211 165 108)(39 222 166 119)(40 233 167 70)(41 184 168 81)(42 195 169 92)(43 206 170 103)(44 217 171 114)(45 228 172 65)(46 239 173 76)(47 190 174 87)(48 201 175 98)(49 212 176 109)(50 223 177 120)(51 234 178 71)(52 185 179 82)(53 196 180 93)(54 207 121 104)(55 218 122 115)(56 229 123 66)(57 240 124 77)(58 191 125 88)(59 202 126 99)(60 213 127 110)

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

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

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

66 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D5A5B6A6B6C8A8B8C8D10A···10F12A12B12C12D15A15B20A20B20C20D20E···20L24A···24H30A···30F60A···60H
order1222223444455666888810···101212121215152020202020···2024···2430···3060···60
size11116060222121222222101010102···2222244444412···1210···104···44···4

66 irreducible representations

dim11111222222222222222244444444
type++++++++++++++++++++++
imageC1C2C2C2C4S3D4D4D5D6D8SD16D10C4×S3C3⋊D4D12C4×D5D20C5⋊D4C24⋊C2D24S3×D5D4⋊D5Q8⋊D5D30.C2C3⋊D20C5⋊D12C5⋊D24Dic6⋊D5
kernelD6015C4C6×C52C8C5×C4⋊Dic3C2×D60D60C2×C52C8C60C2×C30C4⋊Dic3C2×C20C30C30C2×C12C20C20C2×C10C12C12C2×C6C10C10C2×C4C6C6C4C4C22C2C2
# reps11114111212222224444422222244

Matrix representation of D6015C4 in GL5(𝔽241)

10000
09914200
09919800
00051190
000511
,
10000
01429900
01989900
0000240
0002400
,
640000
0669400
02817500
00076192
00049165

G:=sub<GL(5,GF(241))| [1,0,0,0,0,0,99,99,0,0,0,142,198,0,0,0,0,0,51,51,0,0,0,190,1],[1,0,0,0,0,0,142,198,0,0,0,99,99,0,0,0,0,0,0,240,0,0,0,240,0],[64,0,0,0,0,0,66,28,0,0,0,94,175,0,0,0,0,0,76,49,0,0,0,192,165] >;

D6015C4 in GAP, Magma, Sage, TeX

D_{60}\rtimes_{15}C_4
% in TeX

G:=Group("D60:15C4");
// GroupNames label

G:=SmallGroup(480,45);
// by ID

G=gap.SmallGroup(480,45);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,197,64,219,100,1356,18822]);
// Polycyclic

G:=Group<a,b,c|a^60=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^11,c*b*c^-1=a^25*b>;
// generators/relations

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