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G = C604D4order 480 = 25·3·5

4th semidirect product of C60 and D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C604D4, D62D20, (C6×D20)⋊5C2, (C2×D20)⋊4S3, (S3×C10)⋊10D4, C53(D63D4), C33(C207D4), C43(C15⋊D4), C121(C5⋊D4), C605C427C2, C6.21(C2×D20), C2.22(S3×D20), C10.20(S3×D4), C1510(C4⋊D4), C2010(C3⋊D4), (C2×C20).301D6, C30.151(C2×D4), C6.58(C4○D20), C30.90(C4○D4), (C2×C12).130D10, (C22×D5).17D6, D10⋊Dic317C2, (C2×C30).146C23, (C2×C60).120C22, (C22×S3).75D10, C10.33(D42S3), C2.18(D205S3), (C2×Dic3).156D10, (C10×Dic3).190C22, (C2×Dic15).113C22, (S3×C2×C4)⋊1D5, (S3×C2×C20)⋊2C2, (C2×C15⋊D4)⋊6C2, C6.87(C2×C5⋊D4), (C2×C4).111(S3×D5), C10.88(C2×C3⋊D4), C2.20(C2×C15⋊D4), (D5×C2×C6).31C22, C22.198(C2×S3×D5), (S3×C2×C10).90C22, (C2×C6).158(C22×D5), (C2×C10).158(C22×S3), SmallGroup(480,532)

Series: Derived Chief Lower central Upper central

C1C2×C30 — C604D4
C1C5C15C30C2×C30D5×C2×C6C2×C15⋊D4 — C604D4
C15C2×C30 — C604D4
C1C22C2×C4

Generators and relations for C604D4
 G = < a,b,c | a60=b4=c2=1, bab-1=a-1, cac=a19, cbc=b-1 >

Subgroups: 988 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 [×3], C12 [×2], D6 [×2], D6 [×2], C2×C6, C2×C6 [×6], 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], C2×Dic3, C2×Dic3 [×2], C3⋊D4 [×4], C2×C12, C3×D4 [×2], C22×S3, C22×C6 [×2], C5×S3 [×2], C3×D5 [×2], C30 [×3], C4⋊D4, D20 [×2], C2×Dic5 [×2], C5⋊D4 [×4], C2×C20, C2×C20 [×3], C22×D5 [×2], C22×C10, C4⋊Dic3, C6.D4 [×2], S3×C2×C4, C2×C3⋊D4 [×2], C6×D4, C5×Dic3, Dic15 [×2], C60 [×2], C6×D5 [×6], S3×C10 [×2], S3×C10 [×2], C2×C30, C4⋊Dic5, D10⋊C4 [×2], C2×D20, C2×C5⋊D4 [×2], C22×C20, D63D4, C15⋊D4 [×4], C3×D20 [×2], S3×C20 [×2], C10×Dic3, C2×Dic15 [×2], C2×C60, D5×C2×C6 [×2], S3×C2×C10, C207D4, D10⋊Dic3 [×2], C605C4, C2×C15⋊D4 [×2], C6×D20, S3×C2×C20, C604D4
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, C4⋊D4, D20 [×2], C5⋊D4 [×2], C22×D5, S3×D4, D42S3, C2×C3⋊D4, S3×D5, C2×D20, C4○D20, C2×C5⋊D4, D63D4, C15⋊D4 [×2], C2×S3×D5, C207D4, D205S3, S3×D20, C2×C15⋊D4, C604D4

Smallest permutation representation of C604D4
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 67 219 175)(2 66 220 174)(3 65 221 173)(4 64 222 172)(5 63 223 171)(6 62 224 170)(7 61 225 169)(8 120 226 168)(9 119 227 167)(10 118 228 166)(11 117 229 165)(12 116 230 164)(13 115 231 163)(14 114 232 162)(15 113 233 161)(16 112 234 160)(17 111 235 159)(18 110 236 158)(19 109 237 157)(20 108 238 156)(21 107 239 155)(22 106 240 154)(23 105 181 153)(24 104 182 152)(25 103 183 151)(26 102 184 150)(27 101 185 149)(28 100 186 148)(29 99 187 147)(30 98 188 146)(31 97 189 145)(32 96 190 144)(33 95 191 143)(34 94 192 142)(35 93 193 141)(36 92 194 140)(37 91 195 139)(38 90 196 138)(39 89 197 137)(40 88 198 136)(41 87 199 135)(42 86 200 134)(43 85 201 133)(44 84 202 132)(45 83 203 131)(46 82 204 130)(47 81 205 129)(48 80 206 128)(49 79 207 127)(50 78 208 126)(51 77 209 125)(52 76 210 124)(53 75 211 123)(54 74 212 122)(55 73 213 121)(56 72 214 180)(57 71 215 179)(58 70 216 178)(59 69 217 177)(60 68 218 176)
(1 175)(2 134)(3 153)(4 172)(5 131)(6 150)(7 169)(8 128)(9 147)(10 166)(11 125)(12 144)(13 163)(14 122)(15 141)(16 160)(17 179)(18 138)(19 157)(20 176)(21 135)(22 154)(23 173)(24 132)(25 151)(26 170)(27 129)(28 148)(29 167)(30 126)(31 145)(32 164)(33 123)(34 142)(35 161)(36 180)(37 139)(38 158)(39 177)(40 136)(41 155)(42 174)(43 133)(44 152)(45 171)(46 130)(47 149)(48 168)(49 127)(50 146)(51 165)(52 124)(53 143)(54 162)(55 121)(56 140)(57 159)(58 178)(59 137)(60 156)(61 225)(62 184)(63 203)(64 222)(65 181)(66 200)(67 219)(68 238)(69 197)(70 216)(71 235)(72 194)(73 213)(74 232)(75 191)(76 210)(77 229)(78 188)(79 207)(80 226)(81 185)(82 204)(83 223)(84 182)(85 201)(86 220)(87 239)(88 198)(89 217)(90 236)(91 195)(92 214)(93 233)(94 192)(95 211)(96 230)(97 189)(98 208)(99 227)(100 186)(101 205)(102 224)(103 183)(104 202)(105 221)(106 240)(107 199)(108 218)(109 237)(110 196)(111 215)(112 234)(113 193)(114 212)(115 231)(116 190)(117 209)(118 228)(119 187)(120 206)

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

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

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

72 conjugacy classes

class 1 2A2B2C2D2E2F2G 3 4A4B4C4D4E4F5A5B6A6B6C6D6E6F6G10A···10F10G···10N12A12B15A15B20A···20H20I···20P30A···30F60A···60H
order12222222344444455666666610···1010···101212151520···2020···2030···3060···60
size111166202022266606022222202020202···26···644442···26···64···44···4

72 irreducible representations

dim111111222222222222224444444
type+++++++++++++++++-+-+-+
imageC1C2C2C2C2C2S3D4D4D5D6D6C4○D4D10D10D10C3⋊D4C5⋊D4D20C4○D20S3×D4D42S3S3×D5C15⋊D4C2×S3×D5D205S3S3×D20
kernelC604D4D10⋊Dic3C605C4C2×C15⋊D4C6×D20S3×C2×C20C2×D20C60S3×C10S3×C2×C4C2×C20C22×D5C30C2×Dic3C2×C12C22×S3C20C12D6C6C10C10C2×C4C4C22C2C2
# reps121211122212222248881124244

Matrix representation of C604D4 in GL6(𝔽61)

3300000
0370000
0006000
001100
00004118
000003
,
010000
6000000
001000
00606000
00003958
00006022
,
010000
100000
001000
000100
0000225
0000139

G:=sub<GL(6,GF(61))| [33,0,0,0,0,0,0,37,0,0,0,0,0,0,0,1,0,0,0,0,60,1,0,0,0,0,0,0,41,0,0,0,0,0,18,3],[0,60,0,0,0,0,1,0,0,0,0,0,0,0,1,60,0,0,0,0,0,60,0,0,0,0,0,0,39,60,0,0,0,0,58,22],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,22,1,0,0,0,0,5,39] >;

C604D4 in GAP, Magma, Sage, TeX

C_{60}\rtimes_4D_4
% in TeX

G:=Group("C60:4D4");
// GroupNames label

G:=SmallGroup(480,532);
// by ID

G=gap.SmallGroup(480,532);
# by ID

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

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

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