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G = D6012C4order 480 = 25·3·5

6th semidirect product of D60 and C4 acting via C4/C2=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D6012C4, C6.11D40, C60.33D4, C30.17D8, C20.49D12, C30.5SD16, C4⋊Dic51S3, C12.3(C4×D5), C20.63(C4×S3), C60.91(C2×C4), (C2×C30).15D4, (C2×C6).30D20, C52(C6.D8), C157(D4⋊C4), C31(D205C4), C10.6(D4⋊S3), (C2×D60).12C2, (C2×C12).50D10, (C2×C20).280D6, C6.4(C40⋊C2), C2.2(C3⋊D40), C10.19(D6⋊C4), C4.14(C5⋊D12), C12.13(C5⋊D4), C4.1(D30.C2), (C2×C60).99C22, C6.4(D10⋊C4), C2.5(D304C4), C30.51(C22⋊C4), C10.1(Q82S3), C2.1(C15⋊SD16), C22.13(C3⋊D20), (C2×C3⋊C8)⋊2D5, (C10×C3⋊C8)⋊5C2, (C3×C4⋊Dic5)⋊1C2, (C2×C4).83(S3×D5), (C2×C10).24(C3⋊D4), SmallGroup(480,44)

Series: Derived Chief Lower central Upper central

C1C60 — D6012C4
C1C5C15C30C2×C30C2×C60C3×C4⋊Dic5 — D6012C4
C15C30C60 — D6012C4
C1C22C2×C4

Generators and relations for D6012C4
 G = < a,b,c | a60=b2=c4=1, bab=a-1, cac-1=a19, cbc-1=a33b >

Subgroups: 812 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], C12 [×2], C12, D6 [×4], C2×C6, C15, C4⋊C4, C2×C8, C2×D4, Dic5, C20 [×2], D10 [×4], C2×C10, C3⋊C8, D12 [×3], C2×C12, C2×C12, C22×S3, D15 [×2], C30 [×3], D4⋊C4, C40, D20 [×3], C2×Dic5, C2×C20, C22×D5, C2×C3⋊C8, C3×C4⋊C4, C2×D12, C3×Dic5, C60 [×2], D30 [×4], C2×C30, C4⋊Dic5, C2×C40, C2×D20, C6.D8, C5×C3⋊C8, C6×Dic5, D60 [×2], D60, C2×C60, C22×D15, D205C4, C3×C4⋊Dic5, C10×C3⋊C8, C2×D60, D6012C4
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, D6⋊C4, D4⋊S3, Q82S3, S3×D5, C40⋊C2, D40, D10⋊C4, C6.D8, D30.C2, C3⋊D20, C5⋊D12, D205C4, C3⋊D40, C15⋊SD16, D304C4, D6012C4

Smallest permutation representation of D6012C4
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 60)(2 59)(3 58)(4 57)(5 56)(6 55)(7 54)(8 53)(9 52)(10 51)(11 50)(12 49)(13 48)(14 47)(15 46)(16 45)(17 44)(18 43)(19 42)(20 41)(21 40)(22 39)(23 38)(24 37)(25 36)(26 35)(27 34)(28 33)(29 32)(30 31)(61 64)(62 63)(65 120)(66 119)(67 118)(68 117)(69 116)(70 115)(71 114)(72 113)(73 112)(74 111)(75 110)(76 109)(77 108)(78 107)(79 106)(80 105)(81 104)(82 103)(83 102)(84 101)(85 100)(86 99)(87 98)(88 97)(89 96)(90 95)(91 94)(92 93)(121 165)(122 164)(123 163)(124 162)(125 161)(126 160)(127 159)(128 158)(129 157)(130 156)(131 155)(132 154)(133 153)(134 152)(135 151)(136 150)(137 149)(138 148)(139 147)(140 146)(141 145)(142 144)(166 180)(167 179)(168 178)(169 177)(170 176)(171 175)(172 174)(182 240)(183 239)(184 238)(185 237)(186 236)(187 235)(188 234)(189 233)(190 232)(191 231)(192 230)(193 229)(194 228)(195 227)(196 226)(197 225)(198 224)(199 223)(200 222)(201 221)(202 220)(203 219)(204 218)(205 217)(206 216)(207 215)(208 214)(209 213)(210 212)
(1 204 93 166)(2 223 94 125)(3 182 95 144)(4 201 96 163)(5 220 97 122)(6 239 98 141)(7 198 99 160)(8 217 100 179)(9 236 101 138)(10 195 102 157)(11 214 103 176)(12 233 104 135)(13 192 105 154)(14 211 106 173)(15 230 107 132)(16 189 108 151)(17 208 109 170)(18 227 110 129)(19 186 111 148)(20 205 112 167)(21 224 113 126)(22 183 114 145)(23 202 115 164)(24 221 116 123)(25 240 117 142)(26 199 118 161)(27 218 119 180)(28 237 120 139)(29 196 61 158)(30 215 62 177)(31 234 63 136)(32 193 64 155)(33 212 65 174)(34 231 66 133)(35 190 67 152)(36 209 68 171)(37 228 69 130)(38 187 70 149)(39 206 71 168)(40 225 72 127)(41 184 73 146)(42 203 74 165)(43 222 75 124)(44 181 76 143)(45 200 77 162)(46 219 78 121)(47 238 79 140)(48 197 80 159)(49 216 81 178)(50 235 82 137)(51 194 83 156)(52 213 84 175)(53 232 85 134)(54 191 86 153)(55 210 87 172)(56 229 88 131)(57 188 89 150)(58 207 90 169)(59 226 91 128)(60 185 92 147)

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

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

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

72 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D5A5B6A6B6C8A8B8C8D10A···10F12A12B12C12D12E12F15A15B20A···20H30A···30F40A···40P60A···60H
order1222223444455666888810···10121212121212151520···2030···3040···4060···60
size1111606022220202222266662···24420202020442···24···46···64···4

72 irreducible representations

dim11111222222222222222244444444
type++++++++++++++++++++++
imageC1C2C2C2C4S3D4D4D5D6D8SD16D10C4×S3D12C3⋊D4C4×D5C5⋊D4D20C40⋊C2D40D4⋊S3Q82S3S3×D5D30.C2C5⋊D12C3⋊D20C3⋊D40C15⋊SD16
kernelD6012C4C3×C4⋊Dic5C10×C3⋊C8C2×D60D60C4⋊Dic5C60C2×C30C2×C3⋊C8C2×C20C30C30C2×C12C20C20C2×C10C12C12C2×C6C6C6C10C10C2×C4C4C4C22C2C2
# reps11114111212222224448811222244

Matrix representation of D6012C4 in GL6(𝔽241)

220500000
49210000
005218900
005224000
00001187
000067239
,
21680000
1922200000
000100
001000
00002187
000067239
,
102670000
2081390000
001954600
00644600
00002400
00000240

G:=sub<GL(6,GF(241))| [220,49,0,0,0,0,50,21,0,0,0,0,0,0,52,52,0,0,0,0,189,240,0,0,0,0,0,0,1,67,0,0,0,0,187,239],[21,192,0,0,0,0,68,220,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,2,67,0,0,0,0,187,239],[102,208,0,0,0,0,67,139,0,0,0,0,0,0,195,64,0,0,0,0,46,46,0,0,0,0,0,0,240,0,0,0,0,0,0,240] >;

D6012C4 in GAP, Magma, Sage, TeX

D_{60}\rtimes_{12}C_4
% in TeX

G:=Group("D60:12C4");
// GroupNames label

G:=SmallGroup(480,44);
// by ID

G=gap.SmallGroup(480,44);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,204,675,80,1356,18822]);
// Polycyclic

G:=Group<a,b,c|a^60=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^19,c*b*c^-1=a^33*b>;
// generators/relations

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