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## G = C60.17D4order 480 = 25·3·5

### 17th non-split extension by C60 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C30 — C60.17D4
 Chief series C1 — C5 — C15 — C30 — C2×C30 — C2×Dic15 — C4×Dic15 — C60.17D4
 Lower central C15 — C2×C30 — C60.17D4
 Upper central C1 — C22 — C2×D4

Generators and relations for C60.17D4
G = < a,b,c | a60=b4=1, c2=a30, bab-1=a29, cac-1=a-1, cbc-1=a30b-1 >

Subgroups: 692 in 152 conjugacy classes, 55 normal (23 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C5, C6, C6, C6, C2×C4, C2×C4, D4, Q8, C23, C10, C10, C10, Dic3, C12, C2×C6, C2×C6, C15, C42, C22⋊C4, C2×D4, C2×Q8, Dic5, C20, C2×C10, C2×C10, Dic6, C2×Dic3, C2×C12, C3×D4, C22×C6, C30, C30, C30, C4.4D4, Dic10, C2×Dic5, C2×C20, C5×D4, C22×C10, C4×Dic3, C6.D4, C2×Dic6, C6×D4, Dic15, C60, C2×C30, C2×C30, C4×Dic5, C23.D5, C2×Dic10, D4×C10, C23.12D6, Dic30, C2×Dic15, C2×C60, D4×C15, C22×C30, C20.17D4, C4×Dic15, C30.38D4, C2×Dic30, D4×C30, C60.17D4
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, C4○D4, D10, C3⋊D4, C22×S3, D15, C4.4D4, C5⋊D4, C22×D5, D42S3, C2×C3⋊D4, D30, D42D5, C2×C5⋊D4, C23.12D6, C157D4, C22×D15, C20.17D4, D42D15, C2×C157D4, C60.17D4

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

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

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

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

84 conjugacy classes

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

84 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + - - - image C1 C2 C2 C2 C2 S3 D4 D5 D6 D6 C4○D4 D10 D10 C3⋊D4 D15 C5⋊D4 D30 D30 C15⋊7D4 D4⋊2S3 D4⋊2D5 D4⋊2D15 kernel C60.17D4 C4×Dic15 C30.38D4 C2×Dic30 D4×C30 D4×C10 C60 C6×D4 C2×C20 C22×C10 C30 C2×C12 C22×C6 C20 C2×D4 C12 C2×C4 C23 C4 C10 C6 C2 # reps 1 1 4 1 1 1 2 2 1 2 4 2 4 4 4 8 4 8 16 2 4 8

Matrix representation of C60.17D4 in GL6(𝔽61)

 60 5 0 0 0 0 39 48 0 0 0 0 0 0 9 0 0 0 0 0 6 34 0 0 0 0 0 0 60 13 0 0 0 0 28 1
,
 46 36 0 0 0 0 48 15 0 0 0 0 0 0 43 47 0 0 0 0 45 18 0 0 0 0 0 0 11 0 0 0 0 0 0 11
,
 46 36 0 0 0 0 48 15 0 0 0 0 0 0 18 14 0 0 0 0 51 43 0 0 0 0 0 0 50 21 0 0 0 0 0 11

`G:=sub<GL(6,GF(61))| [60,39,0,0,0,0,5,48,0,0,0,0,0,0,9,6,0,0,0,0,0,34,0,0,0,0,0,0,60,28,0,0,0,0,13,1],[46,48,0,0,0,0,36,15,0,0,0,0,0,0,43,45,0,0,0,0,47,18,0,0,0,0,0,0,11,0,0,0,0,0,0,11],[46,48,0,0,0,0,36,15,0,0,0,0,0,0,18,51,0,0,0,0,14,43,0,0,0,0,0,0,50,0,0,0,0,0,21,11] >;`

C60.17D4 in GAP, Magma, Sage, TeX

`C_{60}._{17}D_4`
`% in TeX`

`G:=Group("C60.17D4");`
`// GroupNames label`

`G:=SmallGroup(480,901);`
`// by ID`

`G=gap.SmallGroup(480,901);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,64,590,135,2693,18822]);`
`// Polycyclic`

`G:=Group<a,b,c|a^60=b^4=1,c^2=a^30,b*a*b^-1=a^29,c*a*c^-1=a^-1,c*b*c^-1=a^30*b^-1>;`
`// generators/relations`

׿
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