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## G = C60.10D4order 480 = 25·3·5

### 10th non-split extension by C60 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C30 — C60.10D4
 Chief series C1 — C5 — C15 — C30 — C60 — C2×C60 — C60.7C4 — C60.10D4
 Lower central C15 — C30 — C2×C30 — C60.10D4
 Upper central C1 — C2 — C2×C4 — C2×Q8

Generators and relations for C60.10D4
G = < a,b,c | a60=1, b4=a30, c2=a45, bab-1=a-1, cac-1=a29, cbc-1=a45b3 >

Subgroups: 244 in 76 conjugacy classes, 39 normal (21 characteristic)
C1, C2, C2, C3, C4 [×2], C4 [×2], C22, C5, C6, C6, C8 [×2], C2×C4, C2×C4 [×2], Q8 [×2], C10, C10, C12 [×2], C12 [×2], C2×C6, C15, M4(2) [×2], C2×Q8, C20 [×2], C20 [×2], C2×C10, C3⋊C8 [×2], C2×C12, C2×C12 [×2], C3×Q8 [×2], C30, C30, C4.10D4, C52C8 [×2], C2×C20, C2×C20 [×2], C5×Q8 [×2], C4.Dic3 [×2], C6×Q8, C60 [×2], C60 [×2], C2×C30, C4.Dic5 [×2], Q8×C10, C12.10D4, C153C8 [×2], C2×C60, C2×C60 [×2], Q8×C15 [×2], C20.10D4, C60.7C4 [×2], Q8×C30, C60.10D4
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], D5, Dic3 [×2], D6, C22⋊C4, Dic5 [×2], D10, C2×Dic3, C3⋊D4 [×2], D15, C4.10D4, C2×Dic5, C5⋊D4 [×2], C6.D4, Dic15 [×2], D30, C23.D5, C12.10D4, C2×Dic15, C157D4 [×2], C20.10D4, C30.38D4, C60.10D4

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

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

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

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

81 conjugacy classes

 class 1 2A 2B 3 4A 4B 4C 4D 5A 5B 6A 6B 6C 8A 8B 8C 8D 10A ··· 10F 12A ··· 12F 15A 15B 15C 15D 20A ··· 20L 30A ··· 30L 60A ··· 60X order 1 2 2 3 4 4 4 4 5 5 6 6 6 8 8 8 8 10 ··· 10 12 ··· 12 15 15 15 15 20 ··· 20 30 ··· 30 60 ··· 60 size 1 1 2 2 2 2 4 4 2 2 2 2 2 60 60 60 60 2 ··· 2 4 ··· 4 2 2 2 2 4 ··· 4 2 ··· 2 4 ··· 4

81 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + - + - + + - + - image C1 C2 C2 C4 S3 D4 D5 Dic3 D6 Dic5 D10 C3⋊D4 D15 C5⋊D4 Dic15 D30 C15⋊7D4 C4.10D4 C12.10D4 C20.10D4 C60.10D4 kernel C60.10D4 C60.7C4 Q8×C30 C2×C60 Q8×C10 C60 C6×Q8 C2×C20 C2×C20 C2×C12 C2×C12 C20 C2×Q8 C12 C2×C4 C2×C4 C4 C15 C5 C3 C1 # reps 1 2 1 4 1 2 2 2 1 4 2 4 4 8 8 4 16 1 2 4 8

Matrix representation of C60.10D4 in GL6(𝔽241)

 15 0 0 0 0 0 0 225 0 0 0 0 0 0 0 91 0 0 0 0 150 0 0 0 0 0 0 0 0 98 0 0 0 0 143 0
,
 0 197 0 0 0 0 126 0 0 0 0 0 0 0 0 0 50 106 0 0 0 0 106 191 0 0 135 50 0 0 0 0 50 106 0 0
,
 0 197 0 0 0 0 115 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 240 0 0 0

`G:=sub<GL(6,GF(241))| [15,0,0,0,0,0,0,225,0,0,0,0,0,0,0,150,0,0,0,0,91,0,0,0,0,0,0,0,0,143,0,0,0,0,98,0],[0,126,0,0,0,0,197,0,0,0,0,0,0,0,0,0,135,50,0,0,0,0,50,106,0,0,50,106,0,0,0,0,106,191,0,0],[0,115,0,0,0,0,197,0,0,0,0,0,0,0,0,0,0,240,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0] >;`

C60.10D4 in GAP, Magma, Sage, TeX

`C_{60}._{10}D_4`
`% in TeX`

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

`G:=SmallGroup(480,196);`
`// by ID`

`G=gap.SmallGroup(480,196);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,120,219,100,675,2693,18822]);`
`// Polycyclic`

`G:=Group<a,b,c|a^60=1,b^4=a^30,c^2=a^45,b*a*b^-1=a^-1,c*a*c^-1=a^29,c*b*c^-1=a^45*b^3>;`
`// generators/relations`

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