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## G = C60⋊10D4order 480 = 25·3·5

### 10th semidirect product of C60 and D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C30 — C60⋊10D4
 Chief series C1 — C5 — C15 — C30 — C2×C30 — D5×C2×C6 — C2×C15⋊D4 — C60⋊10D4
 Lower central C15 — C2×C30 — C60⋊10D4
 Upper central C1 — C22 — C2×C4

Generators and relations for C6010D4
G = < a,b,c | a60=b4=c2=1, bab-1=a29, cac=a19, cbc=b-1 >

Subgroups: 1244 in 216 conjugacy classes, 56 normal (22 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 [×4], D5 [×2], C10, C10 [×2], C10 [×2], Dic3 [×4], C12 [×2], D6 [×6], C2×C6, C2×C6 [×6], C15, C42, C2×D4 [×6], 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], C5×S3 [×2], C3×D5 [×2], C30, 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], C6×D5 [×6], S3×C10 [×6], C2×C30, C4×Dic5, C2×D20, C2×C5⋊D4 [×4], D4×C10, C123D4, C15⋊D4 [×8], C3×D20 [×2], C5×D12 [×2], C2×Dic15 [×2], C2×C60, D5×C2×C6 [×2], S3×C2×C10 [×2], C20⋊D4, C4×Dic15, C2×C15⋊D4 [×4], C6×D20, C10×D12, C6010D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×6], C23, D5, D6 [×3], C2×D4 [×3], D10 [×3], C3⋊D4 [×2], C22×S3, C41D4, C5⋊D4 [×2], C22×D5, S3×D4 [×2], C2×C3⋊D4, S3×D5, D4×D5 [×2], C2×C5⋊D4, C123D4, C15⋊D4 [×2], C2×S3×D5, C20⋊D4, C20⋊D6 [×2], C2×C15⋊D4, C6010D4

Smallest permutation representation of C6010D4
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)
(1 169 206 87)(2 138 207 116)(3 167 208 85)(4 136 209 114)(5 165 210 83)(6 134 211 112)(7 163 212 81)(8 132 213 110)(9 161 214 79)(10 130 215 108)(11 159 216 77)(12 128 217 106)(13 157 218 75)(14 126 219 104)(15 155 220 73)(16 124 221 102)(17 153 222 71)(18 122 223 100)(19 151 224 69)(20 180 225 98)(21 149 226 67)(22 178 227 96)(23 147 228 65)(24 176 229 94)(25 145 230 63)(26 174 231 92)(27 143 232 61)(28 172 233 90)(29 141 234 119)(30 170 235 88)(31 139 236 117)(32 168 237 86)(33 137 238 115)(34 166 239 84)(35 135 240 113)(36 164 181 82)(37 133 182 111)(38 162 183 80)(39 131 184 109)(40 160 185 78)(41 129 186 107)(42 158 187 76)(43 127 188 105)(44 156 189 74)(45 125 190 103)(46 154 191 72)(47 123 192 101)(48 152 193 70)(49 121 194 99)(50 150 195 68)(51 179 196 97)(52 148 197 66)(53 177 198 95)(54 146 199 64)(55 175 200 93)(56 144 201 62)(57 173 202 91)(58 142 203 120)(59 171 204 89)(60 140 205 118)
(1 16)(2 35)(3 54)(4 13)(5 32)(6 51)(7 10)(8 29)(9 48)(11 26)(12 45)(14 23)(15 42)(17 20)(18 39)(19 58)(21 36)(22 55)(24 33)(25 52)(27 30)(28 49)(31 46)(34 43)(37 40)(38 59)(41 56)(44 53)(47 50)(57 60)(61 170)(62 129)(63 148)(64 167)(65 126)(66 145)(67 164)(68 123)(69 142)(70 161)(71 180)(72 139)(73 158)(74 177)(75 136)(76 155)(77 174)(78 133)(79 152)(80 171)(81 130)(82 149)(83 168)(84 127)(85 146)(86 165)(87 124)(88 143)(89 162)(90 121)(91 140)(92 159)(93 178)(94 137)(95 156)(96 175)(97 134)(98 153)(99 172)(100 131)(101 150)(102 169)(103 128)(104 147)(105 166)(106 125)(107 144)(108 163)(109 122)(110 141)(111 160)(112 179)(113 138)(114 157)(115 176)(116 135)(117 154)(118 173)(119 132)(120 151)(181 226)(182 185)(183 204)(184 223)(186 201)(187 220)(188 239)(189 198)(190 217)(191 236)(192 195)(193 214)(194 233)(196 211)(197 230)(199 208)(200 227)(202 205)(203 224)(206 221)(207 240)(209 218)(210 237)(212 215)(213 234)(216 231)(219 228)(222 225)(229 238)(232 235)

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

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

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

60 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 4C 4D 4E 4F 5A 5B 6A 6B 6C 6D 6E 6F 6G 10A ··· 10F 10G ··· 10N 12A 12B 15A 15B 20A 20B 20C 20D 30A ··· 30F 60A ··· 60H order 1 2 2 2 2 2 2 2 3 4 4 4 4 4 4 5 5 6 6 6 6 6 6 6 10 ··· 10 10 ··· 10 12 12 15 15 20 20 20 20 30 ··· 30 60 ··· 60 size 1 1 1 1 12 12 20 20 2 2 2 30 30 30 30 2 2 2 2 2 20 20 20 20 2 ··· 2 12 ··· 12 4 4 4 4 4 4 4 4 4 ··· 4 4 ··· 4

60 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + + + + + + + + + + + - + image C1 C2 C2 C2 C2 S3 D4 D4 D5 D6 D6 D10 D10 C3⋊D4 C5⋊D4 S3×D4 S3×D5 D4×D5 C15⋊D4 C2×S3×D5 C20⋊D6 kernel C60⋊10D4 C4×Dic15 C2×C15⋊D4 C6×D20 C10×D12 C2×D20 Dic15 C60 C2×D12 C2×C20 C22×D5 C2×C12 C22×S3 C20 C12 C10 C2×C4 C6 C4 C22 C2 # reps 1 1 4 1 1 1 4 2 2 1 2 2 4 4 8 2 2 4 4 2 8

Matrix representation of C6010D4 in GL6(𝔽61)

 18 18 0 0 0 0 43 60 0 0 0 0 0 0 60 15 0 0 0 0 12 2 0 0 0 0 0 0 0 1 0 0 0 0 60 0
,
 31 17 0 0 0 0 8 30 0 0 0 0 0 0 31 48 0 0 0 0 41 30 0 0 0 0 0 0 0 1 0 0 0 0 60 0
,
 1 0 0 0 0 0 43 60 0 0 0 0 0 0 60 0 0 0 0 0 0 60 0 0 0 0 0 0 0 1 0 0 0 0 1 0

G:=sub<GL(6,GF(61))| [18,43,0,0,0,0,18,60,0,0,0,0,0,0,60,12,0,0,0,0,15,2,0,0,0,0,0,0,0,60,0,0,0,0,1,0],[31,8,0,0,0,0,17,30,0,0,0,0,0,0,31,41,0,0,0,0,48,30,0,0,0,0,0,0,0,60,0,0,0,0,1,0],[1,43,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

C6010D4 in GAP, Magma, Sage, TeX

C_{60}\rtimes_{10}D_4
% in TeX

G:=Group("C60:10D4");
// GroupNames label

G:=SmallGroup(480,539);
// by ID

G=gap.SmallGroup(480,539);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,141,422,219,100,1356,18822]);
// Polycyclic

G:=Group<a,b,c|a^60=b^4=c^2=1,b*a*b^-1=a^29,c*a*c=a^19,c*b*c=b^-1>;
// generators/relations

׿
×
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