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## G = D60.C4order 480 = 25·3·5

### 2nd non-split extension by D60 of C4 acting faithfully

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — D60.C4
 Chief series C1 — C5 — C15 — C30 — C3×Dic5 — C3×C5⋊C8 — D15⋊C8 — D60.C4
 Lower central C15 — C30 — D60.C4
 Upper central C1 — C2 — C4

Generators and relations for D60.C4
G = < a,b,c | a60=b2=1, c4=a30, bab=a-1, cac-1=a13, cbc-1=a12b >

Subgroups: 612 in 124 conjugacy classes, 46 normal (30 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C6, C8, C2×C4, D4, Q8, D5, C10, Dic3, C12, C12, D6, C2×C6, C15, C2×C8, M4(2), C4○D4, Dic5, C20, C20, D10, D10, C3⋊C8, C24, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C3×D5, D15, C30, C8○D4, C5⋊C8, C5⋊C8, C4×D5, C4×D5, D20, C5×Q8, S3×C8, C8⋊S3, C4.Dic3, C2×C24, C4○D12, C5×Dic3, C3×Dic5, C60, C6×D5, D30, D5⋊C8, D5⋊C8, C4.F5, Q82D5, C8○D12, C3×C5⋊C8, C15⋊C8, D30.C2, C3⋊D20, D5×C12, C5×Dic6, D60, Q8.F5, D15⋊C8, Dic3.F5, C3×D5⋊C8, C12.F5, C12.28D10, D60.C4
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, C22×C4, F5, C4×S3, C22×S3, C8○D4, C2×F5, S3×C2×C4, C22×F5, C8○D12, S3×F5, Q8.F5, C2×S3×F5, D60.C4

Smallest permutation representation of D60.C4
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 60)(2 59)(3 58)(4 57)(5 56)(6 55)(7 54)(8 53)(9 52)(10 51)(11 50)(12 49)(13 48)(14 47)(15 46)(16 45)(17 44)(18 43)(19 42)(20 41)(21 40)(22 39)(23 38)(24 37)(25 36)(26 35)(27 34)(28 33)(29 32)(30 31)(61 100)(62 99)(63 98)(64 97)(65 96)(66 95)(67 94)(68 93)(69 92)(70 91)(71 90)(72 89)(73 88)(74 87)(75 86)(76 85)(77 84)(78 83)(79 82)(80 81)(101 120)(102 119)(103 118)(104 117)(105 116)(106 115)(107 114)(108 113)(109 112)(110 111)(121 166)(122 165)(123 164)(124 163)(125 162)(126 161)(127 160)(128 159)(129 158)(130 157)(131 156)(132 155)(133 154)(134 153)(135 152)(136 151)(137 150)(138 149)(139 148)(140 147)(141 146)(142 145)(143 144)(167 180)(168 179)(169 178)(170 177)(171 176)(172 175)(173 174)(181 184)(182 183)(185 240)(186 239)(187 238)(188 237)(189 236)(190 235)(191 234)(192 233)(193 232)(194 231)(195 230)(196 229)(197 228)(198 227)(199 226)(200 225)(201 224)(202 223)(203 222)(204 221)(205 220)(206 219)(207 218)(208 217)(209 216)(210 215)(211 214)(212 213)
(1 213 144 81 31 183 174 111)(2 190 133 94 32 220 163 64)(3 227 122 107 33 197 152 77)(4 204 171 120 34 234 141 90)(5 181 160 73 35 211 130 103)(6 218 149 86 36 188 179 116)(7 195 138 99 37 225 168 69)(8 232 127 112 38 202 157 82)(9 209 176 65 39 239 146 95)(10 186 165 78 40 216 135 108)(11 223 154 91 41 193 124 61)(12 200 143 104 42 230 173 74)(13 237 132 117 43 207 162 87)(14 214 121 70 44 184 151 100)(15 191 170 83 45 221 140 113)(16 228 159 96 46 198 129 66)(17 205 148 109 47 235 178 79)(18 182 137 62 48 212 167 92)(19 219 126 75 49 189 156 105)(20 196 175 88 50 226 145 118)(21 233 164 101 51 203 134 71)(22 210 153 114 52 240 123 84)(23 187 142 67 53 217 172 97)(24 224 131 80 54 194 161 110)(25 201 180 93 55 231 150 63)(26 238 169 106 56 208 139 76)(27 215 158 119 57 185 128 89)(28 192 147 72 58 222 177 102)(29 229 136 85 59 199 166 115)(30 206 125 98 60 236 155 68)```

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

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

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

45 conjugacy classes

 class 1 2A 2B 2C 2D 3 4A 4B 4C 4D 4E 5 6A 6B 6C 8A 8B 8C 8D 8E 8F 8G 8H 8I 8J 10 12A 12B 12C 12D 15 20A 20B 20C 24A ··· 24H 30 60A 60B order 1 2 2 2 2 3 4 4 4 4 4 5 6 6 6 8 8 8 8 8 8 8 8 8 8 10 12 12 12 12 15 20 20 20 24 ··· 24 30 60 60 size 1 1 10 30 30 2 2 5 5 6 6 4 2 10 10 5 5 5 5 10 10 30 30 30 30 4 2 2 10 10 8 8 24 24 10 ··· 10 8 8 8

45 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 8 8 8 8 type + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 C4 C4 S3 D6 D6 C4×S3 C4×S3 C8○D4 C8○D12 F5 C2×F5 C2×F5 S3×F5 Q8.F5 C2×S3×F5 D60.C4 kernel D60.C4 D15⋊C8 Dic3.F5 C3×D5⋊C8 C12.F5 C12.28D10 C3⋊D20 C5×Dic6 D60 D5⋊C8 C5⋊C8 C4×D5 C20 D10 C15 C5 Dic6 Dic3 C12 C4 C3 C2 C1 # reps 1 2 2 1 1 1 4 2 2 1 2 1 2 2 4 8 1 2 1 1 1 1 2

Matrix representation of D60.C4 in GL8(𝔽241)

 240 239 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 240 1 0 0 0 0 0 0 0 0 0 0 1 240 0 0 0 0 0 0 1 0 0 0 0 0 240 0 1 0 0 0 0 0 0 240 1 0
,
 240 239 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 240 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 240 1 0 0 0 0 0 240 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 240
,
 30 0 0 0 0 0 0 0 0 30 0 0 0 0 0 0 0 0 64 0 0 0 0 0 0 0 0 64 0 0 0 0 0 0 0 0 34 207 224 0 0 0 0 0 17 207 0 34 0 0 0 0 34 0 207 17 0 0 0 0 0 224 207 34

`G:=sub<GL(8,GF(241))| [240,1,0,0,0,0,0,0,239,1,0,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,1,1,1,1,0,0,0,0,240,0,0,0],[240,0,0,0,0,0,0,0,239,1,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,0,1,1,1,1,0,0,0,0,0,0,0,240],[30,0,0,0,0,0,0,0,0,30,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,34,17,34,0,0,0,0,0,207,207,0,224,0,0,0,0,224,0,207,207,0,0,0,0,0,34,17,34] >;`

D60.C4 in GAP, Magma, Sage, TeX

`D_{60}.C_4`
`% in TeX`

`G:=Group("D60.C4");`
`// GroupNames label`

`G:=SmallGroup(480,990);`
`// by ID`

`G=gap.SmallGroup(480,990);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,120,219,100,80,1356,9414,2379]);`
`// Polycyclic`

`G:=Group<a,b,c|a^60=b^2=1,c^4=a^30,b*a*b=a^-1,c*a*c^-1=a^13,c*b*c^-1=a^12*b>;`
`// generators/relations`

׿
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