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G = C6029D4order 480 = 25·3·5

1st semidirect product of C60 and D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C6029D4, C222D60, C23.29D30, (C2×C6)⋊5D20, (C2×D60)⋊9C2, (C2×C10)⋊8D12, (C2×C30)⋊23D4, C43(C157D4), C35(C207D4), C55(C127D4), C605C415C2, (C2×C4).86D30, C6.45(C2×D20), C2.17(C2×D60), (C22×C4)⋊6D15, (C22×C12)⋊6D5, D303C43C2, C2013(C3⋊D4), C1213(C5⋊D4), C1531(C4⋊D4), (C22×C60)⋊10C2, (C22×C20)⋊10S3, C30.273(C2×D4), (C2×C20).395D6, C10.46(C2×D12), (C2×C12).385D10, C6.107(C4○D20), C30.179(C4○D4), (C2×C60).467C22, (C2×C30).304C23, (C22×C6).121D10, (C22×C10).139D6, C10.107(C4○D12), C2.19(D6011C2), C22.56(C22×D15), (C22×C30).144C22, (C2×Dic15).15C22, (C22×D15).11C22, (C2×C157D4)⋊3C2, C2.7(C2×C157D4), C6.102(C2×C5⋊D4), C10.102(C2×C3⋊D4), (C2×C6).300(C22×D5), (C2×C10).299(C22×S3), SmallGroup(480,895)

Series: Derived Chief Lower central Upper central

C1C2×C30 — C6029D4
C1C5C15C30C2×C30C22×D15C2×D60 — C6029D4
C15C2×C30 — C6029D4
C1C22C22×C4

Generators and relations for C6029D4
 G = < a,b,c | a60=b4=c2=1, bab-1=cac=a-1, cbc=b-1 >

Subgroups: 1236 in 188 conjugacy classes, 63 normal (39 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×2], C4 [×3], C22, C22 [×2], C22 [×8], C5, S3 [×2], C6 [×3], C6 [×2], C2×C4 [×2], C2×C4 [×4], D4 [×6], C23, C23 [×2], D5 [×2], C10 [×3], C10 [×2], Dic3 [×2], C12 [×2], C12, D6 [×6], C2×C6, C2×C6 [×2], C2×C6 [×2], C15, C22⋊C4 [×2], C4⋊C4, C22×C4, C2×D4 [×3], Dic5 [×2], C20 [×2], C20, D10 [×6], C2×C10, C2×C10 [×2], C2×C10 [×2], D12 [×2], C2×Dic3 [×2], C3⋊D4 [×4], C2×C12 [×2], C2×C12 [×2], C22×S3 [×2], C22×C6, D15 [×2], C30 [×3], C30 [×2], C4⋊D4, D20 [×2], C2×Dic5 [×2], C5⋊D4 [×4], C2×C20 [×2], C2×C20 [×2], C22×D5 [×2], C22×C10, C4⋊Dic3, D6⋊C4 [×2], C2×D12, C2×C3⋊D4 [×2], C22×C12, Dic15 [×2], C60 [×2], C60, D30 [×6], C2×C30, C2×C30 [×2], C2×C30 [×2], C4⋊Dic5, D10⋊C4 [×2], C2×D20, C2×C5⋊D4 [×2], C22×C20, C127D4, D60 [×2], C2×Dic15 [×2], C157D4 [×4], C2×C60 [×2], C2×C60 [×2], C22×D15 [×2], C22×C30, C207D4, C605C4, D303C4 [×2], C2×D60, C2×C157D4 [×2], C22×C60, C6029D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×4], C23, D5, D6 [×3], C2×D4 [×2], C4○D4, D10 [×3], D12 [×2], C3⋊D4 [×2], C22×S3, D15, C4⋊D4, D20 [×2], C5⋊D4 [×2], C22×D5, C2×D12, C4○D12, C2×C3⋊D4, D30 [×3], C2×D20, C4○D20, C2×C5⋊D4, C127D4, D60 [×2], C157D4 [×2], C22×D15, C207D4, C2×D60, D6011C2, C2×C157D4, C6029D4

Smallest permutation representation of C6029D4
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)
(1 122 198 116)(2 121 199 115)(3 180 200 114)(4 179 201 113)(5 178 202 112)(6 177 203 111)(7 176 204 110)(8 175 205 109)(9 174 206 108)(10 173 207 107)(11 172 208 106)(12 171 209 105)(13 170 210 104)(14 169 211 103)(15 168 212 102)(16 167 213 101)(17 166 214 100)(18 165 215 99)(19 164 216 98)(20 163 217 97)(21 162 218 96)(22 161 219 95)(23 160 220 94)(24 159 221 93)(25 158 222 92)(26 157 223 91)(27 156 224 90)(28 155 225 89)(29 154 226 88)(30 153 227 87)(31 152 228 86)(32 151 229 85)(33 150 230 84)(34 149 231 83)(35 148 232 82)(36 147 233 81)(37 146 234 80)(38 145 235 79)(39 144 236 78)(40 143 237 77)(41 142 238 76)(42 141 239 75)(43 140 240 74)(44 139 181 73)(45 138 182 72)(46 137 183 71)(47 136 184 70)(48 135 185 69)(49 134 186 68)(50 133 187 67)(51 132 188 66)(52 131 189 65)(53 130 190 64)(54 129 191 63)(55 128 192 62)(56 127 193 61)(57 126 194 120)(58 125 195 119)(59 124 196 118)(60 123 197 117)
(1 116)(2 115)(3 114)(4 113)(5 112)(6 111)(7 110)(8 109)(9 108)(10 107)(11 106)(12 105)(13 104)(14 103)(15 102)(16 101)(17 100)(18 99)(19 98)(20 97)(21 96)(22 95)(23 94)(24 93)(25 92)(26 91)(27 90)(28 89)(29 88)(30 87)(31 86)(32 85)(33 84)(34 83)(35 82)(36 81)(37 80)(38 79)(39 78)(40 77)(41 76)(42 75)(43 74)(44 73)(45 72)(46 71)(47 70)(48 69)(49 68)(50 67)(51 66)(52 65)(53 64)(54 63)(55 62)(56 61)(57 120)(58 119)(59 118)(60 117)(121 199)(122 198)(123 197)(124 196)(125 195)(126 194)(127 193)(128 192)(129 191)(130 190)(131 189)(132 188)(133 187)(134 186)(135 185)(136 184)(137 183)(138 182)(139 181)(140 240)(141 239)(142 238)(143 237)(144 236)(145 235)(146 234)(147 233)(148 232)(149 231)(150 230)(151 229)(152 228)(153 227)(154 226)(155 225)(156 224)(157 223)(158 222)(159 221)(160 220)(161 219)(162 218)(163 217)(164 216)(165 215)(166 214)(167 213)(168 212)(169 211)(170 210)(171 209)(172 208)(173 207)(174 206)(175 205)(176 204)(177 203)(178 202)(179 201)(180 200)

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

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

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

126 conjugacy classes

class 1 2A2B2C2D2E2F2G 3 4A4B4C4D4E4F5A5B6A···6G10A···10N12A···12H15A15B15C15D20A···20P30A···30AB60A···60AF
order122222223444444556···610···1012···121515151520···2030···3060···60
size1111226060222226060222···22···22···222222···22···22···2

126 irreducible representations

dim111111222222222222222222222
type++++++++++++++++++++
imageC1C2C2C2C2C2S3D4D4D5D6D6C4○D4D10D10C3⋊D4D12D15C5⋊D4D20C4○D12D30D30C4○D20C157D4D60D6011C2
kernelC6029D4C605C4D303C4C2×D60C2×C157D4C22×C60C22×C20C60C2×C30C22×C12C2×C20C22×C10C30C2×C12C22×C6C20C2×C10C22×C4C12C2×C6C10C2×C4C23C6C4C22C2
# reps112121122221242444884848161616

Matrix representation of C6029D4 in GL4(𝔽61) generated by

462300
382300
00330
005137
,
92200
135200
00418
004920
,
92200
135200
00418
003420
G:=sub<GL(4,GF(61))| [46,38,0,0,23,23,0,0,0,0,33,51,0,0,0,37],[9,13,0,0,22,52,0,0,0,0,41,49,0,0,8,20],[9,13,0,0,22,52,0,0,0,0,41,34,0,0,8,20] >;

C6029D4 in GAP, Magma, Sage, TeX

C_{60}\rtimes_{29}D_4
% in TeX

G:=Group("C60:29D4");
// GroupNames label

G:=SmallGroup(480,895);
// by ID

G=gap.SmallGroup(480,895);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,253,120,254,2693,18822]);
// Polycyclic

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

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