Copied to
clipboard

## G = C60.44D4order 480 = 25·3·5

### 44th non-split extension by C60 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C30 — C60.44D4
 Chief series C1 — C5 — C15 — C30 — C2×C30 — D5×C2×C6 — D10⋊Dic3 — C60.44D4
 Lower central C15 — C2×C30 — C60.44D4
 Upper central C1 — C22 — C2×C4

Generators and relations for C60.44D4
G = < a,b,c | a60=b4=1, c2=a30, bab-1=a41, cac-1=a-1, cbc-1=a30b-1 >

Subgroups: 796 in 152 conjugacy classes, 52 normal (22 characteristic)
C1, C2, C2 [×2], C2 [×2], C3, C4 [×2], C4 [×4], C22, C22 [×6], C5, C6, C6 [×2], C6 [×2], C2×C4, C2×C4 [×4], D4 [×2], Q8 [×2], C23 [×2], D5 [×2], C10, C10 [×2], Dic3 [×4], C12 [×2], C2×C6, C2×C6 [×6], C15, C42, C22⋊C4 [×4], C2×D4, C2×Q8, Dic5 [×2], C20 [×2], C20 [×2], D10 [×6], C2×C10, Dic6 [×2], C2×Dic3 [×2], C2×Dic3 [×2], C2×C12, C3×D4 [×2], C22×C6 [×2], C3×D5 [×2], C30, C30 [×2], C4.4D4, Dic10 [×2], D20 [×2], C2×Dic5 [×2], C2×C20, C2×C20 [×2], C22×D5 [×2], C4×Dic3, C6.D4 [×4], C2×Dic6, C6×D4, C5×Dic3 [×2], Dic15 [×2], C60 [×2], C6×D5 [×6], C2×C30, D10⋊C4 [×4], C4×C20, C2×Dic10, C2×D20, C23.12D6, C3×D20 [×2], C10×Dic3 [×2], Dic30 [×2], C2×Dic15 [×2], C2×C60, D5×C2×C6 [×2], C4.D20, D10⋊Dic3 [×4], Dic3×C20, C6×D20, C2×Dic30, C60.44D4
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, C4.4D4, D20 [×2], C22×D5, D42S3 [×2], C2×C3⋊D4, S3×D5, C2×D20, C4○D20 [×2], C23.12D6, C3⋊D20 [×2], C2×S3×D5, C4.D20, D205S3 [×2], C2×C3⋊D20, C60.44D4

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

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

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

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

72 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 12A 12B 15A 15B 20A ··· 20H 20I ··· 20X 30A ··· 30F 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 12 12 15 15 20 ··· 20 20 ··· 20 30 ··· 30 60 ··· 60 size 1 1 1 1 20 20 2 2 2 6 6 6 6 60 60 2 2 2 2 2 20 20 20 20 2 ··· 2 4 4 4 4 2 ··· 2 6 ··· 6 4 ··· 4 4 ··· 4

72 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + + + + + + - + + + - image C1 C2 C2 C2 C2 S3 D4 D5 D6 D6 C4○D4 D10 D10 C3⋊D4 D20 C4○D20 D4⋊2S3 S3×D5 C3⋊D20 C2×S3×D5 D20⋊5S3 kernel C60.44D4 D10⋊Dic3 Dic3×C20 C6×D20 C2×Dic30 C2×D20 C60 C4×Dic3 C2×C20 C22×D5 C30 C2×Dic3 C2×C12 C20 C12 C6 C10 C2×C4 C4 C22 C2 # reps 1 4 1 1 1 1 2 2 1 2 4 4 2 4 8 16 2 2 4 2 8

Matrix representation of C60.44D4 in GL4(𝔽61) generated by

 48 0 0 0 0 14 0 0 0 0 57 2 0 0 23 34
,
 0 1 0 0 60 0 0 0 0 0 30 45 0 0 60 31
,
 0 1 0 0 1 0 0 0 0 0 46 11 0 0 46 15
`G:=sub<GL(4,GF(61))| [48,0,0,0,0,14,0,0,0,0,57,23,0,0,2,34],[0,60,0,0,1,0,0,0,0,0,30,60,0,0,45,31],[0,1,0,0,1,0,0,0,0,0,46,46,0,0,11,15] >;`

C60.44D4 in GAP, Magma, Sage, TeX

`C_{60}._{44}D_4`
`% in TeX`

`G:=Group("C60.44D4");`
`// GroupNames label`

`G:=SmallGroup(480,440);`
`// by ID`

`G=gap.SmallGroup(480,440);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,141,176,422,100,1356,18822]);`
`// Polycyclic`

`G:=Group<a,b,c|a^60=b^4=1,c^2=a^30,b*a*b^-1=a^41,c*a*c^-1=a^-1,c*b*c^-1=a^30*b^-1>;`
`// generators/relations`

׿
×
𝔽