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G = D608C4order 480 = 25·3·5

2nd semidirect product of D60 and C4 acting via C4/C2=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D608C4, C6.5D40, C2.2D120, C10.5D24, C30.26D8, C60.202D4, C30.18SD16, C22.10D60, (C2×C40)⋊4S3, (C2×C8)⋊2D15, (C2×C24)⋊4D5, (C2×C120)⋊6C2, C605C41C2, C4.8(C4×D15), C20.83(C4×S3), (C2×D60).1C2, (C2×C6).16D20, C12.51(C4×D5), (C2×C4).74D30, C53(C2.D24), C32(D205C4), C60.188(C2×C4), (C2×C10).16D12, (C2×C20).387D6, (C2×C30).102D4, C6.3(C40⋊C2), C1514(D4⋊C4), C10.3(C24⋊C2), C2.3(C24⋊D5), C10.33(D6⋊C4), (C2×C12).389D10, C20.99(C3⋊D4), C4.20(C157D4), C12.99(C5⋊D4), C2.8(D303C4), C30.75(C22⋊C4), (C2×C60).474C22, C6.18(D10⋊C4), SmallGroup(480,181)

Series: Derived Chief Lower central Upper central

C1C60 — D608C4
C1C5C15C30C60C2×C60C2×D60 — D608C4
C15C30C60 — D608C4
C1C22C2×C4C2×C8

Generators and relations for D608C4
 G = < a,b,c | a60=b2=c4=1, bab=cac-1=a-1, cbc-1=a43b >

Subgroups: 884 in 100 conjugacy classes, 41 normal (39 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, Dic5, C20 [×2], D10 [×4], C2×C10, C24, D12 [×3], C2×Dic3, C2×C12, C22×S3, D15 [×2], C30 [×3], D4⋊C4, C40, D20 [×3], C2×Dic5, C2×C20, C22×D5, C4⋊Dic3, C2×C24, C2×D12, Dic15, C60 [×2], D30 [×4], C2×C30, C4⋊Dic5, C2×C40, C2×D20, C2.D24, C120, D60 [×2], D60, C2×Dic15, C2×C60, C22×D15, D205C4, C605C4, C2×C120, C2×D60, D608C4
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], D5, D6, C22⋊C4, D8, SD16, D10, C4×S3, D12, C3⋊D4, D15, D4⋊C4, C4×D5, D20, C5⋊D4, C24⋊C2, D24, D6⋊C4, D30, C40⋊C2, D40, D10⋊C4, C2.D24, C4×D15, D60, C157D4, D205C4, C24⋊D5, D120, D303C4, D608C4

Smallest permutation representation of D608C4
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 74)(62 73)(63 72)(64 71)(65 70)(66 69)(67 68)(75 120)(76 119)(77 118)(78 117)(79 116)(80 115)(81 114)(82 113)(83 112)(84 111)(85 110)(86 109)(87 108)(88 107)(89 106)(90 105)(91 104)(92 103)(93 102)(94 101)(95 100)(96 99)(97 98)(121 123)(124 180)(125 179)(126 178)(127 177)(128 176)(129 175)(130 174)(131 173)(132 172)(133 171)(134 170)(135 169)(136 168)(137 167)(138 166)(139 165)(140 164)(141 163)(142 162)(143 161)(144 160)(145 159)(146 158)(147 157)(148 156)(149 155)(150 154)(151 153)(181 195)(182 194)(183 193)(184 192)(185 191)(186 190)(187 189)(196 240)(197 239)(198 238)(199 237)(200 236)(201 235)(202 234)(203 233)(204 232)(205 231)(206 230)(207 229)(208 228)(209 227)(210 226)(211 225)(212 224)(213 223)(214 222)(215 221)(216 220)(217 219)
(1 226 98 130)(2 225 99 129)(3 224 100 128)(4 223 101 127)(5 222 102 126)(6 221 103 125)(7 220 104 124)(8 219 105 123)(9 218 106 122)(10 217 107 121)(11 216 108 180)(12 215 109 179)(13 214 110 178)(14 213 111 177)(15 212 112 176)(16 211 113 175)(17 210 114 174)(18 209 115 173)(19 208 116 172)(20 207 117 171)(21 206 118 170)(22 205 119 169)(23 204 120 168)(24 203 61 167)(25 202 62 166)(26 201 63 165)(27 200 64 164)(28 199 65 163)(29 198 66 162)(30 197 67 161)(31 196 68 160)(32 195 69 159)(33 194 70 158)(34 193 71 157)(35 192 72 156)(36 191 73 155)(37 190 74 154)(38 189 75 153)(39 188 76 152)(40 187 77 151)(41 186 78 150)(42 185 79 149)(43 184 80 148)(44 183 81 147)(45 182 82 146)(46 181 83 145)(47 240 84 144)(48 239 85 143)(49 238 86 142)(50 237 87 141)(51 236 88 140)(52 235 89 139)(53 234 90 138)(54 233 91 137)(55 232 92 136)(56 231 93 135)(57 230 94 134)(58 229 95 133)(59 228 96 132)(60 227 97 131)

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

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

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

126 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D5A5B6A6B6C8A8B8C8D10A···10F12A12B12C12D15A15B15C15D20A···20H24A···24H30A···30L40A···40P60A···60P120A···120AF
order1222223444455666888810···10121212121515151520···2024···2430···3040···4060···60120···120
size1111606022260602222222222···2222222222···22···22···22···22···22···2

126 irreducible representations

dim111112222222222222222222222222
type+++++++++++++++++++
imageC1C2C2C2C4S3D4D4D5D6D8SD16D10C4×S3C3⋊D4D12D15C4×D5C5⋊D4D20C24⋊C2D24D30C40⋊C2D40C4×D15C157D4D60C24⋊D5D120
kernelD608C4C605C4C2×C120C2×D60D60C2×C40C60C2×C30C2×C24C2×C20C30C30C2×C12C20C20C2×C10C2×C8C12C12C2×C6C10C10C2×C4C6C6C4C4C22C2C2
# reps11114111212222224444444888881616

Matrix representation of D608C4 in GL5(𝔽241)

10000
01429900
01424300
00080131
000110147
,
2400000
01429900
01989900
000157148
0009484
,
1770000
0669400
02817500
0007878
000197163

G:=sub<GL(5,GF(241))| [1,0,0,0,0,0,142,142,0,0,0,99,43,0,0,0,0,0,80,110,0,0,0,131,147],[240,0,0,0,0,0,142,198,0,0,0,99,99,0,0,0,0,0,157,94,0,0,0,148,84],[177,0,0,0,0,0,66,28,0,0,0,94,175,0,0,0,0,0,78,197,0,0,0,78,163] >;

D608C4 in GAP, Magma, Sage, TeX

D_{60}\rtimes_8C_4
% in TeX

G:=Group("D60:8C4");
// GroupNames label

G:=SmallGroup(480,181);
// by ID

G=gap.SmallGroup(480,181);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,85,92,422,100,2693,18822]);
// Polycyclic

G:=Group<a,b,c|a^60=b^2=c^4=1,b*a*b=c*a*c^-1=a^-1,c*b*c^-1=a^43*b>;
// generators/relations

׿
×
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