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G = C606D4order 480 = 25·3·5

6th semidirect product of C60 and D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C606D4, D63D20, C207D12, C4⋊Dic59S3, (S3×C10)⋊11D4, (C2×D60)⋊22C2, C53(C12⋊D4), C122(C5⋊D4), C42(C5⋊D12), C31(C207D4), C2.26(S3×D20), C6.24(C2×D20), C10.23(S3×D4), C1512(C4⋊D4), C30.154(C2×D4), (C2×C20).303D6, C10.62(C2×D12), D304C421C2, C30.92(C4○D4), C6.14(C4○D20), (C2×C12).132D10, (C2×Dic5).46D6, (C2×C30).150C23, (C2×C60).122C22, (C22×S3).76D10, C2.19(D60⋊C2), C10.19(Q83S3), (C2×Dic3).158D10, (C6×Dic5).90C22, (C22×D15).53C22, (C10×Dic3).192C22, (S3×C2×C4)⋊2D5, (S3×C2×C20)⋊3C2, (C2×C5⋊D12)⋊6C2, (C3×C4⋊Dic5)⋊6C2, C6.17(C2×C5⋊D4), (C2×C4).113(S3×D5), C2.20(C2×C5⋊D12), C22.202(C2×S3×D5), (S3×C2×C10).91C22, (C2×C6).162(C22×D5), (C2×C10).162(C22×S3), SmallGroup(480,536)

Series: Derived Chief Lower central Upper central

C1C2×C30 — C606D4
C1C5C15C30C2×C30C6×Dic5C2×C5⋊D12 — C606D4
C15C2×C30 — C606D4
C1C22C2×C4

Generators and relations for C606D4
 G = < a,b,c | a60=b4=c2=1, bab-1=a19, cac=a-1, cbc=b-1 >

Subgroups: 1228 in 188 conjugacy classes, 56 normal (34 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×2], C4 [×3], C22, C22 [×10], C5, S3 [×4], C6 [×3], C2×C4, C2×C4 [×5], D4 [×6], C23 [×3], D5 [×2], C10 [×3], C10 [×2], Dic3, C12 [×2], C12 [×2], D6 [×2], D6 [×8], C2×C6, C15, C22⋊C4 [×2], C4⋊C4, C22×C4, C2×D4 [×3], Dic5 [×2], C20 [×2], C20, D10 [×6], C2×C10, C2×C10 [×4], C4×S3 [×2], D12 [×6], C2×Dic3, C2×C12, C2×C12 [×2], C22×S3, C22×S3 [×2], C5×S3 [×2], D15 [×2], C30 [×3], C4⋊D4, D20 [×2], C2×Dic5 [×2], C5⋊D4 [×4], C2×C20, C2×C20 [×3], C22×D5 [×2], C22×C10, D6⋊C4 [×2], C3×C4⋊C4, S3×C2×C4, C2×D12 [×3], C5×Dic3, C3×Dic5 [×2], C60 [×2], S3×C10 [×2], S3×C10 [×2], D30 [×6], C2×C30, C4⋊Dic5, D10⋊C4 [×2], C2×D20, C2×C5⋊D4 [×2], C22×C20, C12⋊D4, C5⋊D12 [×4], C6×Dic5 [×2], S3×C20 [×2], C10×Dic3, D60 [×2], C2×C60, S3×C2×C10, C22×D15 [×2], C207D4, D304C4 [×2], C3×C4⋊Dic5, C2×C5⋊D12 [×2], S3×C2×C20, C2×D60, C606D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×4], C23, D5, D6 [×3], C2×D4 [×2], C4○D4, D10 [×3], D12 [×2], C22×S3, C4⋊D4, D20 [×2], C5⋊D4 [×2], C22×D5, C2×D12, S3×D4, Q83S3, S3×D5, C2×D20, C4○D20, C2×C5⋊D4, C12⋊D4, C5⋊D12 [×2], C2×S3×D5, C207D4, D60⋊C2, S3×D20, C2×C5⋊D12, C606D4

Smallest permutation representation of C606D4
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)
(1 182 168 66)(2 201 169 85)(3 220 170 104)(4 239 171 63)(5 198 172 82)(6 217 173 101)(7 236 174 120)(8 195 175 79)(9 214 176 98)(10 233 177 117)(11 192 178 76)(12 211 179 95)(13 230 180 114)(14 189 121 73)(15 208 122 92)(16 227 123 111)(17 186 124 70)(18 205 125 89)(19 224 126 108)(20 183 127 67)(21 202 128 86)(22 221 129 105)(23 240 130 64)(24 199 131 83)(25 218 132 102)(26 237 133 61)(27 196 134 80)(28 215 135 99)(29 234 136 118)(30 193 137 77)(31 212 138 96)(32 231 139 115)(33 190 140 74)(34 209 141 93)(35 228 142 112)(36 187 143 71)(37 206 144 90)(38 225 145 109)(39 184 146 68)(40 203 147 87)(41 222 148 106)(42 181 149 65)(43 200 150 84)(44 219 151 103)(45 238 152 62)(46 197 153 81)(47 216 154 100)(48 235 155 119)(49 194 156 78)(50 213 157 97)(51 232 158 116)(52 191 159 75)(53 210 160 94)(54 229 161 113)(55 188 162 72)(56 207 163 91)(57 226 164 110)(58 185 165 69)(59 204 166 88)(60 223 167 107)
(1 66)(2 65)(3 64)(4 63)(5 62)(6 61)(7 120)(8 119)(9 118)(10 117)(11 116)(12 115)(13 114)(14 113)(15 112)(16 111)(17 110)(18 109)(19 108)(20 107)(21 106)(22 105)(23 104)(24 103)(25 102)(26 101)(27 100)(28 99)(29 98)(30 97)(31 96)(32 95)(33 94)(34 93)(35 92)(36 91)(37 90)(38 89)(39 88)(40 87)(41 86)(42 85)(43 84)(44 83)(45 82)(46 81)(47 80)(48 79)(49 78)(50 77)(51 76)(52 75)(53 74)(54 73)(55 72)(56 71)(57 70)(58 69)(59 68)(60 67)(121 229)(122 228)(123 227)(124 226)(125 225)(126 224)(127 223)(128 222)(129 221)(130 220)(131 219)(132 218)(133 217)(134 216)(135 215)(136 214)(137 213)(138 212)(139 211)(140 210)(141 209)(142 208)(143 207)(144 206)(145 205)(146 204)(147 203)(148 202)(149 201)(150 200)(151 199)(152 198)(153 197)(154 196)(155 195)(156 194)(157 193)(158 192)(159 191)(160 190)(161 189)(162 188)(163 187)(164 186)(165 185)(166 184)(167 183)(168 182)(169 181)(170 240)(171 239)(172 238)(173 237)(174 236)(175 235)(176 234)(177 233)(178 232)(179 231)(180 230)

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

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

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

72 conjugacy classes

class 1 2A2B2C2D2E2F2G 3 4A4B4C4D4E4F5A5B6A6B6C10A···10F10G···10N12A12B12C12D12E12F15A15B20A···20H20I···20P30A···30F60A···60H
order1222222234444445566610···1010···10121212121212151520···2020···2030···3060···60
size1111666060222662020222222···26···64420202020442···26···64···44···4

72 irreducible representations

dim111111222222222222224444444
type++++++++++++++++++++++++
imageC1C2C2C2C2C2S3D4D4D5D6D6C4○D4D10D10D10D12C5⋊D4D20C4○D20S3×D4Q83S3S3×D5C5⋊D12C2×S3×D5D60⋊C2S3×D20
kernelC606D4D304C4C3×C4⋊Dic5C2×C5⋊D12S3×C2×C20C2×D60C4⋊Dic5C60S3×C10S3×C2×C4C2×Dic5C2×C20C30C2×Dic3C2×C12C22×S3C20C12D6C6C10C10C2×C4C4C22C2C2
# reps121211122221222248881124244

Matrix representation of C606D4 in GL6(𝔽61)

58380000
0200000
0059200
00593200
00006014
0000392
,
1410000
47470000
001000
00176000
0000600
0000060
,
47340000
14140000
001000
00176000
0000600
0000391

G:=sub<GL(6,GF(61))| [58,0,0,0,0,0,38,20,0,0,0,0,0,0,59,59,0,0,0,0,2,32,0,0,0,0,0,0,60,39,0,0,0,0,14,2],[14,47,0,0,0,0,1,47,0,0,0,0,0,0,1,17,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[47,14,0,0,0,0,34,14,0,0,0,0,0,0,1,17,0,0,0,0,0,60,0,0,0,0,0,0,60,39,0,0,0,0,0,1] >;

C606D4 in GAP, Magma, Sage, TeX

C_{60}\rtimes_6D_4
% in TeX

G:=Group("C60:6D4");
// GroupNames label

G:=SmallGroup(480,536);
// by ID

G=gap.SmallGroup(480,536);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,141,422,100,1356,18822]);
// Polycyclic

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

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