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G = C60⋊D4order 480 = 25·3·5

7th semidirect product of C60 and D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C607D4, D102D12, (C2×D12)⋊3D5, (C6×D5)⋊10D4, C6.22(D4×D5), (C10×D12)⋊5C2, C33(C202D4), C201(C3⋊D4), C42(C15⋊D4), C53(C127D4), C156(C4⋊D4), C605C433C2, C2.24(D5×D12), C30.59(C2×D4), C1210(C5⋊D4), D6⋊Dic516C2, (C2×C20).129D6, C10.23(C2×D12), C30.84(C4○D4), (C2×C12).305D10, (C22×D5).91D6, C10.60(C4○D12), C6.29(D42D5), (C2×C30).139C23, (C2×C60).149C22, (C2×Dic5).183D6, (C22×S3).16D10, C2.16(D125D5), (C6×Dic5).210C22, (C2×Dic15).108C22, (C2×C4×D5)⋊1S3, (D5×C2×C12)⋊2C2, (C2×C15⋊D4)⋊3C2, C6.86(C2×C5⋊D4), (C2×C4).162(S3×D5), C10.87(C2×C3⋊D4), C2.19(C2×C15⋊D4), C22.191(C2×S3×D5), (S3×C2×C10).31C22, (D5×C2×C6).107C22, (C2×C6).151(C22×D5), (C2×C10).151(C22×S3), SmallGroup(480,525)

Series: Derived Chief Lower central Upper central

C1C2×C30 — C60⋊D4
C1C5C15C30C2×C30D5×C2×C6C2×C15⋊D4 — C60⋊D4
C15C2×C30 — C60⋊D4
C1C22C2×C4

Generators and relations for C60⋊D4
 G = < a,b,c | a60=b4=c2=1, bab-1=a-1, cac=a49, cbc=b-1 >

Subgroups: 956 in 188 conjugacy classes, 56 normal (34 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 [×3], D5 [×2], C10 [×3], C10 [×2], Dic3 [×2], C12 [×2], C12, D6 [×6], C2×C6, C2×C6 [×4], C15, C22⋊C4 [×2], C4⋊C4, C22×C4, C2×D4 [×3], Dic5 [×3], C20 [×2], D10 [×2], D10 [×2], C2×C10, C2×C10 [×6], D12 [×2], C2×Dic3 [×2], C3⋊D4 [×4], C2×C12, C2×C12 [×3], C22×S3 [×2], C22×C6, C5×S3 [×2], C3×D5 [×2], C30 [×3], C4⋊D4, C4×D5 [×2], C2×Dic5, C2×Dic5 [×2], C5⋊D4 [×4], C2×C20, C5×D4 [×2], C22×D5, C22×C10 [×2], C4⋊Dic3, D6⋊C4 [×2], C2×D12, C2×C3⋊D4 [×2], C22×C12, C3×Dic5, Dic15 [×2], C60 [×2], C6×D5 [×2], C6×D5 [×2], S3×C10 [×6], C2×C30, C4⋊Dic5, C23.D5 [×2], C2×C4×D5, C2×C5⋊D4 [×2], D4×C10, C127D4, C15⋊D4 [×4], D5×C12 [×2], C6×Dic5, C5×D12 [×2], C2×Dic15 [×2], C2×C60, D5×C2×C6, S3×C2×C10 [×2], C202D4, D6⋊Dic5 [×2], C605C4, C2×C15⋊D4 [×2], D5×C2×C12, C10×D12, C60⋊D4
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, C4⋊D4, C5⋊D4 [×2], C22×D5, C2×D12, C4○D12, C2×C3⋊D4, S3×D5, D4×D5, D42D5, C2×C5⋊D4, C127D4, C15⋊D4 [×2], C2×S3×D5, C202D4, D125D5, D5×D12, C2×C15⋊D4, C60⋊D4

Smallest permutation representation of C60⋊D4
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)
(1 124 67 202)(2 123 68 201)(3 122 69 200)(4 121 70 199)(5 180 71 198)(6 179 72 197)(7 178 73 196)(8 177 74 195)(9 176 75 194)(10 175 76 193)(11 174 77 192)(12 173 78 191)(13 172 79 190)(14 171 80 189)(15 170 81 188)(16 169 82 187)(17 168 83 186)(18 167 84 185)(19 166 85 184)(20 165 86 183)(21 164 87 182)(22 163 88 181)(23 162 89 240)(24 161 90 239)(25 160 91 238)(26 159 92 237)(27 158 93 236)(28 157 94 235)(29 156 95 234)(30 155 96 233)(31 154 97 232)(32 153 98 231)(33 152 99 230)(34 151 100 229)(35 150 101 228)(36 149 102 227)(37 148 103 226)(38 147 104 225)(39 146 105 224)(40 145 106 223)(41 144 107 222)(42 143 108 221)(43 142 109 220)(44 141 110 219)(45 140 111 218)(46 139 112 217)(47 138 113 216)(48 137 114 215)(49 136 115 214)(50 135 116 213)(51 134 117 212)(52 133 118 211)(53 132 119 210)(54 131 120 209)(55 130 61 208)(56 129 62 207)(57 128 63 206)(58 127 64 205)(59 126 65 204)(60 125 66 203)
(2 50)(3 39)(4 28)(5 17)(7 55)(8 44)(9 33)(10 22)(12 60)(13 49)(14 38)(15 27)(18 54)(19 43)(20 32)(23 59)(24 48)(25 37)(29 53)(30 42)(34 58)(35 47)(40 52)(45 57)(61 73)(63 111)(64 100)(65 89)(66 78)(68 116)(69 105)(70 94)(71 83)(74 110)(75 99)(76 88)(79 115)(80 104)(81 93)(84 120)(85 109)(86 98)(90 114)(91 103)(95 119)(96 108)(101 113)(106 118)(121 235)(122 224)(123 213)(124 202)(125 191)(126 240)(127 229)(128 218)(129 207)(130 196)(131 185)(132 234)(133 223)(134 212)(135 201)(136 190)(137 239)(138 228)(139 217)(140 206)(141 195)(142 184)(143 233)(144 222)(145 211)(146 200)(147 189)(148 238)(149 227)(150 216)(151 205)(152 194)(153 183)(154 232)(155 221)(156 210)(157 199)(158 188)(159 237)(160 226)(161 215)(162 204)(163 193)(164 182)(165 231)(166 220)(167 209)(168 198)(169 187)(170 236)(171 225)(172 214)(173 203)(174 192)(175 181)(176 230)(177 219)(178 208)(179 197)(180 186)

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

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

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

66 conjugacy classes

class 1 2A2B2C2D2E2F2G 3 4A4B4C4D4E4F5A5B6A6B6C6D6E6F6G10A···10F10G···10N12A12B12C12D12E12F12G12H15A15B20A20B20C20D30A···30F60A···60H
order12222222344444455666666610···1010···10121212121212121215152020202030···3060···60
size1111101012122221010606022222101010102···212···122222101010104444444···44···4

66 irreducible representations

dim111111222222222222224444444
type++++++++++++++++++--+-+
imageC1C2C2C2C2C2S3D4D4D5D6D6D6C4○D4D10D10C3⋊D4D12C5⋊D4C4○D12S3×D5D4×D5D42D5C15⋊D4C2×S3×D5D125D5D5×D12
kernelC60⋊D4D6⋊Dic5C605C4C2×C15⋊D4D5×C2×C12C10×D12C2×C4×D5C60C6×D5C2×D12C2×Dic5C2×C20C22×D5C30C2×C12C22×S3C20D10C12C10C2×C4C6C6C4C22C2C2
# reps121211122211122444842224244

Matrix representation of C60⋊D4 in GL4(𝔽61) generated by

21000
303200
001818
004360
,
521500
23900
003117
00830
,
1000
506000
0010
004360
G:=sub<GL(4,GF(61))| [21,30,0,0,0,32,0,0,0,0,18,43,0,0,18,60],[52,23,0,0,15,9,0,0,0,0,31,8,0,0,17,30],[1,50,0,0,0,60,0,0,0,0,1,43,0,0,0,60] >;

C60⋊D4 in GAP, Magma, Sage, TeX

C_{60}\rtimes D_4
% in TeX

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

G:=SmallGroup(480,525);
// by ID

G=gap.SmallGroup(480,525);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,141,219,100,1356,18822]);
// Polycyclic

G:=Group<a,b,c|a^60=b^4=c^2=1,b*a*b^-1=a^-1,c*a*c=a^49,c*b*c=b^-1>;
// generators/relations

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