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G = C603D4order 480 = 25·3·5

3rd semidirect product of C60 and D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C603D4, Dic156D4, C23.14D30, (C6×D4)⋊4D5, (C2×D4)⋊6D15, (D4×C10)⋊4S3, (D4×C30)⋊4C2, (C2×D60)⋊12C2, C206(C3⋊D4), C126(C5⋊D4), C54(C123D4), C34(C20⋊D4), C158(C41D4), C41(C157D4), (C2×C4).52D30, C6.122(D4×D5), C2.28(D4×D15), (C4×Dic15)⋊6C2, (C2×C20).150D6, C30.327(C2×D4), C10.124(S3×D4), (C2×C12).149D10, (C2×C60).76C22, (C22×C6).67D10, (C22×C10).82D6, (C2×C30).311C23, (C22×C30).23C22, C22.62(C22×D15), (C22×D15).13C22, (C2×Dic15).173C22, (C2×C157D4)⋊7C2, C6.111(C2×C5⋊D4), C2.16(C2×C157D4), C10.111(C2×C3⋊D4), (C2×C6).307(C22×D5), (C2×C10).306(C22×S3), SmallGroup(480,905)

Series: Derived Chief Lower central Upper central

C1C2×C30 — C603D4
C1C5C15C30C2×C30C22×D15C2×D60 — C603D4
C15C2×C30 — C603D4
C1C22C2×D4

Generators and relations for C603D4
 G = < a,b,c | a60=b4=c2=1, bab-1=a29, cac=a-1, cbc=b-1 >

Subgroups: 1460 in 216 conjugacy classes, 59 normal (23 characteristic)
C1, C2, C2 [×2], C2 [×4], C3, C4 [×2], C4 [×4], C22, C22 [×12], C5, S3 [×2], C6, C6 [×2], C6 [×2], C2×C4, C2×C4 [×2], D4 [×12], C23 [×2], C23 [×2], D5 [×2], C10, C10 [×2], C10 [×2], Dic3 [×4], C12 [×2], D6 [×6], C2×C6, C2×C6 [×6], C15, C42, C2×D4, C2×D4 [×5], Dic5 [×4], C20 [×2], D10 [×6], C2×C10, C2×C10 [×6], D12 [×2], C2×Dic3 [×2], C3⋊D4 [×8], C2×C12, C3×D4 [×2], C22×S3 [×2], C22×C6 [×2], D15 [×2], C30, C30 [×2], C30 [×2], C41D4, D20 [×2], C2×Dic5 [×2], C5⋊D4 [×8], C2×C20, C5×D4 [×2], C22×D5 [×2], C22×C10 [×2], C4×Dic3, C2×D12, C2×C3⋊D4 [×4], C6×D4, Dic15 [×4], C60 [×2], D30 [×6], C2×C30, C2×C30 [×6], C4×Dic5, C2×D20, C2×C5⋊D4 [×4], D4×C10, C123D4, D60 [×2], C2×Dic15 [×2], C157D4 [×8], C2×C60, D4×C15 [×2], C22×D15 [×2], C22×C30 [×2], C20⋊D4, C4×Dic15, C2×D60, C2×C157D4 [×4], D4×C30, C603D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×6], C23, D5, D6 [×3], C2×D4 [×3], D10 [×3], C3⋊D4 [×2], C22×S3, D15, C41D4, C5⋊D4 [×2], C22×D5, S3×D4 [×2], C2×C3⋊D4, D30 [×3], D4×D5 [×2], C2×C5⋊D4, C123D4, C157D4 [×2], C22×D15, C20⋊D4, D4×D15 [×2], C2×C157D4, C603D4

Smallest permutation representation of C603D4
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 91 225 140)(2 120 226 169)(3 89 227 138)(4 118 228 167)(5 87 229 136)(6 116 230 165)(7 85 231 134)(8 114 232 163)(9 83 233 132)(10 112 234 161)(11 81 235 130)(12 110 236 159)(13 79 237 128)(14 108 238 157)(15 77 239 126)(16 106 240 155)(17 75 181 124)(18 104 182 153)(19 73 183 122)(20 102 184 151)(21 71 185 180)(22 100 186 149)(23 69 187 178)(24 98 188 147)(25 67 189 176)(26 96 190 145)(27 65 191 174)(28 94 192 143)(29 63 193 172)(30 92 194 141)(31 61 195 170)(32 90 196 139)(33 119 197 168)(34 88 198 137)(35 117 199 166)(36 86 200 135)(37 115 201 164)(38 84 202 133)(39 113 203 162)(40 82 204 131)(41 111 205 160)(42 80 206 129)(43 109 207 158)(44 78 208 127)(45 107 209 156)(46 76 210 125)(47 105 211 154)(48 74 212 123)(49 103 213 152)(50 72 214 121)(51 101 215 150)(52 70 216 179)(53 99 217 148)(54 68 218 177)(55 97 219 146)(56 66 220 175)(57 95 221 144)(58 64 222 173)(59 93 223 142)(60 62 224 171)
(1 16)(2 15)(3 14)(4 13)(5 12)(6 11)(7 10)(8 9)(17 60)(18 59)(19 58)(20 57)(21 56)(22 55)(23 54)(24 53)(25 52)(26 51)(27 50)(28 49)(29 48)(30 47)(31 46)(32 45)(33 44)(34 43)(35 42)(36 41)(37 40)(38 39)(61 125)(62 124)(63 123)(64 122)(65 121)(66 180)(67 179)(68 178)(69 177)(70 176)(71 175)(72 174)(73 173)(74 172)(75 171)(76 170)(77 169)(78 168)(79 167)(80 166)(81 165)(82 164)(83 163)(84 162)(85 161)(86 160)(87 159)(88 158)(89 157)(90 156)(91 155)(92 154)(93 153)(94 152)(95 151)(96 150)(97 149)(98 148)(99 147)(100 146)(101 145)(102 144)(103 143)(104 142)(105 141)(106 140)(107 139)(108 138)(109 137)(110 136)(111 135)(112 134)(113 133)(114 132)(115 131)(116 130)(117 129)(118 128)(119 127)(120 126)(181 224)(182 223)(183 222)(184 221)(185 220)(186 219)(187 218)(188 217)(189 216)(190 215)(191 214)(192 213)(193 212)(194 211)(195 210)(196 209)(197 208)(198 207)(199 206)(200 205)(201 204)(202 203)(225 240)(226 239)(227 238)(228 237)(229 236)(230 235)(231 234)(232 233)

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

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

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

84 conjugacy classes

class 1 2A2B2C2D2E2F2G 3 4A4B4C4D4E4F5A5B6A6B6C6D6E6F6G10A···10F10G···10N12A12B15A15B15C15D20A20B20C20D30A···30L30M···30AB60A···60H
order12222222344444455666666610···1010···101212151515152020202030···3030···3060···60
size1111446060222303030302222244442···24···444222244442···24···44···4

84 irreducible representations

dim1111122222222222222444
type+++++++++++++++++++
imageC1C2C2C2C2S3D4D4D5D6D6D10D10C3⋊D4D15C5⋊D4D30D30C157D4S3×D4D4×D5D4×D15
kernelC603D4C4×Dic15C2×D60C2×C157D4D4×C30D4×C10Dic15C60C6×D4C2×C20C22×C10C2×C12C22×C6C20C2×D4C12C2×C4C23C4C10C6C2
# reps11141142212244484816248

Matrix representation of C603D4 in GL4(𝔽61) generated by

381400
391600
00140
00360
,
82200
225300
0010
0001
,
181700
424300
00140
00060
G:=sub<GL(4,GF(61))| [38,39,0,0,14,16,0,0,0,0,1,3,0,0,40,60],[8,22,0,0,22,53,0,0,0,0,1,0,0,0,0,1],[18,42,0,0,17,43,0,0,0,0,1,0,0,0,40,60] >;

C603D4 in GAP, Magma, Sage, TeX

C_{60}\rtimes_3D_4
% in TeX

G:=Group("C60:3D4");
// GroupNames label

G:=SmallGroup(480,905);
// by ID

G=gap.SmallGroup(480,905);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,253,120,254,219,2693,18822]);
// Polycyclic

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

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