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## G = D4×C60order 480 = 25·3·5

### Direct product of C60 and D4

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — D4×C60
 Chief series C1 — C2 — C22 — C2×C10 — C2×C30 — C2×C60 — C15×C22⋊C4 — D4×C60
 Lower central C1 — C2 — D4×C60
 Upper central C1 — C2×C60 — D4×C60

Generators and relations for D4×C60
G = < a,b,c | a60=b4=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 248 in 188 conjugacy classes, 128 normal (48 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×4], C4 [×3], C22, C22 [×4], C22 [×4], C5, C6 [×3], C6 [×4], C2×C4 [×3], C2×C4 [×2], C2×C4 [×4], D4 [×4], C23 [×2], C10 [×3], C10 [×4], C12 [×4], C12 [×3], C2×C6, C2×C6 [×4], C2×C6 [×4], C15, C42, C22⋊C4 [×2], C4⋊C4, C22×C4 [×2], C2×D4, C20 [×4], C20 [×3], C2×C10, C2×C10 [×4], C2×C10 [×4], C2×C12 [×3], C2×C12 [×2], C2×C12 [×4], C3×D4 [×4], C22×C6 [×2], C30 [×3], C30 [×4], C4×D4, C2×C20 [×3], C2×C20 [×2], C2×C20 [×4], C5×D4 [×4], C22×C10 [×2], C4×C12, C3×C22⋊C4 [×2], C3×C4⋊C4, C22×C12 [×2], C6×D4, C60 [×4], C60 [×3], C2×C30, C2×C30 [×4], C2×C30 [×4], C4×C20, C5×C22⋊C4 [×2], C5×C4⋊C4, C22×C20 [×2], D4×C10, D4×C12, C2×C60 [×3], C2×C60 [×2], C2×C60 [×4], D4×C15 [×4], C22×C30 [×2], D4×C20, C4×C60, C15×C22⋊C4 [×2], C15×C4⋊C4, C22×C60 [×2], D4×C30, D4×C60
Quotients: C1, C2 [×7], C3, C4 [×4], C22 [×7], C5, C6 [×7], C2×C4 [×6], D4 [×2], C23, C10 [×7], C12 [×4], C2×C6 [×7], C15, C22×C4, C2×D4, C4○D4, C20 [×4], C2×C10 [×7], C2×C12 [×6], C3×D4 [×2], C22×C6, C30 [×7], C4×D4, C2×C20 [×6], C5×D4 [×2], C22×C10, C22×C12, C6×D4, C3×C4○D4, C60 [×4], C2×C30 [×7], C22×C20, D4×C10, C5×C4○D4, D4×C12, C2×C60 [×6], D4×C15 [×2], C22×C30, D4×C20, C22×C60, D4×C30, C15×C4○D4, D4×C60

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

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

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

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

300 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 4A 4B 4C 4D 4E ··· 4L 5A 5B 5C 5D 6A ··· 6F 6G ··· 6N 10A ··· 10L 10M ··· 10AB 12A ··· 12H 12I ··· 12X 15A ··· 15H 20A ··· 20P 20Q ··· 20AV 30A ··· 30X 30Y ··· 30BD 60A ··· 60AF 60AG ··· 60CR order 1 2 2 2 2 2 2 2 3 3 4 4 4 4 4 ··· 4 5 5 5 5 6 ··· 6 6 ··· 6 10 ··· 10 10 ··· 10 12 ··· 12 12 ··· 12 15 ··· 15 20 ··· 20 20 ··· 20 30 ··· 30 30 ··· 30 60 ··· 60 60 ··· 60 size 1 1 1 1 2 2 2 2 1 1 1 1 1 1 2 ··· 2 1 1 1 1 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 1 ··· 1 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2

300 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + + image C1 C2 C2 C2 C2 C2 C3 C4 C5 C6 C6 C6 C6 C6 C10 C10 C10 C10 C10 C12 C15 C20 C30 C30 C30 C30 C30 C60 D4 C4○D4 C3×D4 C5×D4 C3×C4○D4 C5×C4○D4 D4×C15 C15×C4○D4 kernel D4×C60 C4×C60 C15×C22⋊C4 C15×C4⋊C4 C22×C60 D4×C30 D4×C20 D4×C15 D4×C12 C4×C20 C5×C22⋊C4 C5×C4⋊C4 C22×C20 D4×C10 C4×C12 C3×C22⋊C4 C3×C4⋊C4 C22×C12 C6×D4 C5×D4 C4×D4 C3×D4 C42 C22⋊C4 C4⋊C4 C22×C4 C2×D4 D4 C60 C30 C20 C12 C10 C6 C4 C2 # reps 1 1 2 1 2 1 2 8 4 2 4 2 4 2 4 8 4 8 4 16 8 32 8 16 8 16 8 64 2 2 4 8 4 8 16 16

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

 27 0 0 0 0 50 0 0 0 0 32 0 0 0 0 32
,
 1 0 0 0 0 1 0 0 0 0 0 1 0 0 60 0
,
 60 0 0 0 0 60 0 0 0 0 1 0 0 0 0 60
G:=sub<GL(4,GF(61))| [27,0,0,0,0,50,0,0,0,0,32,0,0,0,0,32],[1,0,0,0,0,1,0,0,0,0,0,60,0,0,1,0],[60,0,0,0,0,60,0,0,0,0,1,0,0,0,0,60] >;

D4×C60 in GAP, Magma, Sage, TeX

D_4\times C_{60}
% in TeX

G:=Group("D4xC60");
// GroupNames label

G:=SmallGroup(480,923);
// by ID

G=gap.SmallGroup(480,923);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-5,-2,-2,1680,1709,1276]);
// Polycyclic

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

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