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G = C602D4order 480 = 25·3·5

2nd semidirect product of C60 and D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C602D4, D3011D4, C23.13D30, (C6×D4)⋊3D5, (D4×C30)⋊3C2, (D4×C10)⋊3S3, (C2×D4)⋊4D15, C55(D63D4), C35(C202D4), C205(C3⋊D4), C125(C5⋊D4), C42(C157D4), C605C420C2, C2.26(D4×D15), C6.120(D4×D5), (C2×C4).51D30, C1534(C4⋊D4), (C2×C20).149D6, C30.383(C2×D4), C10.122(S3×D4), (C2×C12).148D10, (C2×C60).75C22, (C22×C10).80D6, (C22×C6).65D10, C30.225(C4○D4), C30.38D411C2, (C2×C30).309C23, C6.104(D42D5), C2.17(D42D15), (C22×C30).21C22, C10.104(D42S3), C22.60(C22×D15), (C2×Dic15).18C22, (C22×D15).88C22, (C2×C4×D15)⋊2C2, (C2×C157D4)⋊5C2, C6.109(C2×C5⋊D4), C2.14(C2×C157D4), C10.109(C2×C3⋊D4), (C2×C6).305(C22×D5), (C2×C10).304(C22×S3), SmallGroup(480,903)

Series: Derived Chief Lower central Upper central

C1C2×C30 — C602D4
C1C5C15C30C2×C30C22×D15C2×C4×D15 — C602D4
C15C2×C30 — C602D4
C1C22C2×D4

Generators and relations for C602D4
 G = < a,b,c | a60=b4=c2=1, bab-1=a-1, cac=a29, cbc=b-1 >

Subgroups: 1076 in 188 conjugacy classes, 57 normal (33 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×2], C4 [×3], C22, C22 [×10], C5, S3 [×2], C6 [×3], C6 [×2], C2×C4, C2×C4 [×5], D4 [×6], C23 [×2], C23, D5 [×2], C10 [×3], C10 [×2], Dic3 [×3], C12 [×2], D6 [×4], C2×C6, C2×C6 [×6], C15, C22⋊C4 [×2], C4⋊C4, C22×C4, C2×D4, C2×D4 [×2], Dic5 [×3], C20 [×2], D10 [×4], C2×C10, C2×C10 [×6], C4×S3 [×2], C2×Dic3 [×3], C3⋊D4 [×4], C2×C12, C3×D4 [×2], C22×S3, C22×C6 [×2], D15 [×2], C30 [×3], C30 [×2], C4⋊D4, C4×D5 [×2], C2×Dic5 [×3], C5⋊D4 [×4], C2×C20, C5×D4 [×2], C22×D5, C22×C10 [×2], C4⋊Dic3, C6.D4 [×2], S3×C2×C4, C2×C3⋊D4 [×2], C6×D4, Dic15 [×3], C60 [×2], D30 [×2], D30 [×2], C2×C30, C2×C30 [×6], C4⋊Dic5, C23.D5 [×2], C2×C4×D5, C2×C5⋊D4 [×2], D4×C10, D63D4, C4×D15 [×2], C2×Dic15, C2×Dic15 [×2], C157D4 [×4], C2×C60, D4×C15 [×2], C22×D15, C22×C30 [×2], C202D4, C605C4, C30.38D4 [×2], C2×C4×D15, C2×C157D4 [×2], D4×C30, C602D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×4], C23, D5, D6 [×3], C2×D4 [×2], C4○D4, D10 [×3], C3⋊D4 [×2], C22×S3, D15, C4⋊D4, C5⋊D4 [×2], C22×D5, S3×D4, D42S3, C2×C3⋊D4, D30 [×3], D4×D5, D42D5, C2×C5⋊D4, D63D4, C157D4 [×2], C22×D15, C202D4, D4×D15, D42D15, C2×C157D4, C602D4

Smallest permutation representation of C602D4
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)
(1 131 106 206)(2 130 107 205)(3 129 108 204)(4 128 109 203)(5 127 110 202)(6 126 111 201)(7 125 112 200)(8 124 113 199)(9 123 114 198)(10 122 115 197)(11 121 116 196)(12 180 117 195)(13 179 118 194)(14 178 119 193)(15 177 120 192)(16 176 61 191)(17 175 62 190)(18 174 63 189)(19 173 64 188)(20 172 65 187)(21 171 66 186)(22 170 67 185)(23 169 68 184)(24 168 69 183)(25 167 70 182)(26 166 71 181)(27 165 72 240)(28 164 73 239)(29 163 74 238)(30 162 75 237)(31 161 76 236)(32 160 77 235)(33 159 78 234)(34 158 79 233)(35 157 80 232)(36 156 81 231)(37 155 82 230)(38 154 83 229)(39 153 84 228)(40 152 85 227)(41 151 86 226)(42 150 87 225)(43 149 88 224)(44 148 89 223)(45 147 90 222)(46 146 91 221)(47 145 92 220)(48 144 93 219)(49 143 94 218)(50 142 95 217)(51 141 96 216)(52 140 97 215)(53 139 98 214)(54 138 99 213)(55 137 100 212)(56 136 101 211)(57 135 102 210)(58 134 103 209)(59 133 104 208)(60 132 105 207)
(2 30)(3 59)(4 28)(5 57)(6 26)(7 55)(8 24)(9 53)(10 22)(11 51)(12 20)(13 49)(14 18)(15 47)(17 45)(19 43)(21 41)(23 39)(25 37)(27 35)(29 33)(32 60)(34 58)(36 56)(38 54)(40 52)(42 50)(44 48)(62 90)(63 119)(64 88)(65 117)(66 86)(67 115)(68 84)(69 113)(70 82)(71 111)(72 80)(73 109)(74 78)(75 107)(77 105)(79 103)(81 101)(83 99)(85 97)(87 95)(89 93)(92 120)(94 118)(96 116)(98 114)(100 112)(102 110)(104 108)(121 216)(122 185)(123 214)(124 183)(125 212)(126 181)(127 210)(128 239)(129 208)(130 237)(131 206)(132 235)(133 204)(134 233)(135 202)(136 231)(137 200)(138 229)(139 198)(140 227)(141 196)(142 225)(143 194)(144 223)(145 192)(146 221)(147 190)(148 219)(149 188)(150 217)(151 186)(152 215)(153 184)(154 213)(155 182)(156 211)(157 240)(158 209)(159 238)(160 207)(161 236)(162 205)(163 234)(164 203)(165 232)(166 201)(167 230)(168 199)(169 228)(170 197)(171 226)(172 195)(173 224)(174 193)(175 222)(176 191)(177 220)(178 189)(179 218)(180 187)

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

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

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

84 conjugacy classes

class 1 2A2B2C2D2E2F2G 3 4A4B4C4D4E4F5A5B6A6B6C6D6E6F6G10A···10F10G···10N12A12B15A15B15C15D20A20B20C20D30A···30L30M···30AB60A···60H
order12222222344444455666666610···1010···101212151515152020202030···3030···3060···60
size1111443030222303060602222244442···24···444222244442···24···44···4

84 irreducible representations

dim111111222222222222222444444
type++++++++++++++++++-+-+-
imageC1C2C2C2C2C2S3D4D4D5D6D6C4○D4D10D10C3⋊D4D15C5⋊D4D30D30C157D4S3×D4D42S3D4×D5D42D5D4×D15D42D15
kernelC602D4C605C4C30.38D4C2×C4×D15C2×C157D4D4×C30D4×C10C60D30C6×D4C2×C20C22×C10C30C2×C12C22×C6C20C2×D4C12C2×C4C23C4C10C10C6C6C2C2
# reps1121211222122244484816112244

Matrix representation of C602D4 in GL6(𝔽61)

44440000
17600000
000100
00606000
0000546
0000127
,
24530000
34370000
001000
00606000
0000152
0000060
,
36280000
30250000
001000
00606000
000010
000001

G:=sub<GL(6,GF(61))| [44,17,0,0,0,0,44,60,0,0,0,0,0,0,0,60,0,0,0,0,1,60,0,0,0,0,0,0,54,12,0,0,0,0,6,7],[24,34,0,0,0,0,53,37,0,0,0,0,0,0,1,60,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,52,60],[36,30,0,0,0,0,28,25,0,0,0,0,0,0,1,60,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C602D4 in GAP, Magma, Sage, TeX

C_{60}\rtimes_2D_4
% in TeX

G:=Group("C60:2D4");
// GroupNames label

G:=SmallGroup(480,903);
// by ID

G=gap.SmallGroup(480,903);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,120,254,219,2693,18822]);
// Polycyclic

G:=Group<a,b,c|a^60=b^4=c^2=1,b*a*b^-1=a^-1,c*a*c=a^29,c*b*c=b^-1>;
// generators/relations

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