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## G = C60.23D4order 480 = 25·3·5

### 23rd non-split extension by C60 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C30 — C60.23D4
 Chief series C1 — C5 — C15 — C30 — C2×C30 — C22×D15 — C2×D60 — C60.23D4
 Lower central C15 — C2×C30 — C60.23D4
 Upper central C1 — C22 — C2×Q8

Generators and relations for C60.23D4
G = < a,b,c | a60=b4=c2=1, bab-1=a29, cac=a-1, cbc=a30b-1 >

Subgroups: 1012 in 152 conjugacy classes, 55 normal (23 characteristic)
C1, C2, C2 [×2], C2 [×2], C3, C4 [×2], C4 [×4], C22, C22 [×6], C5, S3 [×2], C6, C6 [×2], C2×C4, C2×C4 [×2], C2×C4 [×2], D4 [×2], Q8 [×2], C23 [×2], D5 [×2], C10, C10 [×2], Dic3 [×2], C12 [×2], C12 [×2], D6 [×6], C2×C6, C15, C42, C22⋊C4 [×4], C2×D4, C2×Q8, Dic5 [×2], C20 [×2], C20 [×2], D10 [×6], C2×C10, D12 [×2], C2×Dic3 [×2], C2×C12, C2×C12 [×2], C3×Q8 [×2], C22×S3 [×2], D15 [×2], C30, C30 [×2], C4.4D4, D20 [×2], C2×Dic5 [×2], C2×C20, C2×C20 [×2], C5×Q8 [×2], C22×D5 [×2], C4×Dic3, D6⋊C4 [×4], C2×D12, C6×Q8, Dic15 [×2], C60 [×2], C60 [×2], D30 [×6], C2×C30, C4×Dic5, D10⋊C4 [×4], C2×D20, Q8×C10, C12.23D4, D60 [×2], C2×Dic15 [×2], C2×C60, C2×C60 [×2], Q8×C15 [×2], C22×D15 [×2], C20.23D4, C4×Dic15, D303C4 [×4], C2×D60, Q8×C30, C60.23D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, C4○D4 [×2], D10 [×3], C3⋊D4 [×2], C22×S3, D15, C4.4D4, C5⋊D4 [×2], C22×D5, Q83S3 [×2], C2×C3⋊D4, D30 [×3], Q82D5 [×2], C2×C5⋊D4, C12.23D4, C157D4 [×2], C22×D15, C20.23D4, Q83D15 [×2], C2×C157D4, C60.23D4

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

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

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

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

84 conjugacy classes

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

84 irreducible representations

 dim 1 1 1 1 1 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 C4○D4 D10 C3⋊D4 D15 C5⋊D4 D30 C15⋊7D4 Q8⋊3S3 Q8⋊2D5 Q8⋊3D15 kernel C60.23D4 C4×Dic15 D30⋊3C4 C2×D60 Q8×C30 Q8×C10 C60 C6×Q8 C2×C20 C30 C2×C12 C20 C2×Q8 C12 C2×C4 C4 C10 C6 C2 # reps 1 1 4 1 1 1 2 2 3 4 6 4 4 8 12 16 2 4 8

Matrix representation of C60.23D4 in GL6(𝔽61)

 18 18 0 0 0 0 43 60 0 0 0 0 0 0 47 31 0 0 0 0 30 25 0 0 0 0 0 0 60 8 0 0 0 0 15 1
,
 30 44 0 0 0 0 53 31 0 0 0 0 0 0 24 33 0 0 0 0 14 37 0 0 0 0 0 0 50 27 0 0 0 0 43 11
,
 60 0 0 0 0 0 18 1 0 0 0 0 0 0 37 28 0 0 0 0 47 24 0 0 0 0 0 0 1 53 0 0 0 0 0 60

`G:=sub<GL(6,GF(61))| [18,43,0,0,0,0,18,60,0,0,0,0,0,0,47,30,0,0,0,0,31,25,0,0,0,0,0,0,60,15,0,0,0,0,8,1],[30,53,0,0,0,0,44,31,0,0,0,0,0,0,24,14,0,0,0,0,33,37,0,0,0,0,0,0,50,43,0,0,0,0,27,11],[60,18,0,0,0,0,0,1,0,0,0,0,0,0,37,47,0,0,0,0,28,24,0,0,0,0,0,0,1,0,0,0,0,0,53,60] >;`

C60.23D4 in GAP, Magma, Sage, TeX

`C_{60}._{23}D_4`
`% in TeX`

`G:=Group("C60.23D4");`
`// GroupNames label`

`G:=SmallGroup(480,912);`
`// by ID`

`G=gap.SmallGroup(480,912);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,253,120,254,219,100,2693,18822]);`
`// 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=a^30*b^-1>;`
`// generators/relations`

׿
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