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## G = D60.4C4order 480 = 25·3·5

### 2nd non-split extension by D60 of C4 acting via C4/C2=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — D60.4C4
 Chief series C1 — C5 — C15 — C30 — C60 — C3×C5⋊2C8 — D15⋊2C8 — D60.4C4
 Lower central C15 — C30 — D60.4C4
 Upper central C1 — C4 — C2×C4

Generators and relations for D60.4C4
G = < a,b,c | a60=b2=1, c4=a30, bab=a-1, cac-1=a19, cbc-1=a18b >

Subgroups: 572 in 124 conjugacy classes, 52 normal (36 characteristic)
C1, C2, C2 [×3], C3, C4 [×2], C4 [×2], C22, C22 [×2], C5, S3 [×2], C6, C6, C8 [×4], C2×C4, C2×C4 [×2], D4 [×3], Q8, D5 [×2], C10, C10, Dic3 [×2], C12 [×2], D6 [×2], C2×C6, C15, C2×C8 [×3], M4(2) [×3], C4○D4, Dic5 [×2], C20 [×2], D10 [×2], C2×C10, C3⋊C8 [×2], C24 [×2], Dic6, C4×S3 [×2], D12, C3⋊D4 [×2], C2×C12, D15 [×2], C30, C30, C8○D4, C52C8 [×2], C40 [×2], Dic10, C4×D5 [×2], D20, C5⋊D4 [×2], C2×C20, S3×C8 [×2], C8⋊S3 [×2], C2×C3⋊C8, C3×M4(2), C4○D12, Dic15 [×2], C60 [×2], D30 [×2], C2×C30, C8×D5 [×2], C8⋊D5 [×2], C4.Dic5, C2×C40, C4○D20, D12.C4, C5×C3⋊C8 [×2], C3×C52C8 [×2], Dic30, C4×D15 [×2], D60, C157D4 [×2], C2×C60, D20.3C4, D152C8 [×2], D30.5C4 [×2], C3×C4.Dic5, C10×C3⋊C8, D6011C2, D60.4C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], C23, D5, D6 [×3], C22×C4, D10 [×3], C4×S3 [×2], C22×S3, C8○D4, C4×D5 [×2], C22×D5, S3×C2×C4, S3×D5, C2×C4×D5, D12.C4, D30.C2 [×2], C2×S3×D5, D20.3C4, C2×D30.C2, D60.4C4

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

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

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

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

78 conjugacy classes

 class 1 2A 2B 2C 2D 3 4A 4B 4C 4D 4E 5A 5B 6A 6B 8A 8B 8C 8D 8E 8F 8G 8H 8I 8J 10A ··· 10F 12A 12B 12C 15A 15B 20A ··· 20H 24A 24B 24C 24D 30A ··· 30F 40A ··· 40P 60A ··· 60H order 1 2 2 2 2 3 4 4 4 4 4 5 5 6 6 8 8 8 8 8 8 8 8 8 8 10 ··· 10 12 12 12 15 15 20 ··· 20 24 24 24 24 30 ··· 30 40 ··· 40 60 ··· 60 size 1 1 2 30 30 2 1 1 2 30 30 2 2 2 4 3 3 3 3 6 6 10 10 10 10 2 ··· 2 2 2 4 4 4 2 ··· 2 20 20 20 20 4 ··· 4 6 ··· 6 4 ··· 4

78 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 C4 C4 S3 D5 D6 D6 D10 D10 C4×S3 C4×S3 C8○D4 C4×D5 C4×D5 D20.3C4 S3×D5 D12.C4 D30.C2 C2×S3×D5 D30.C2 D60.4C4 kernel D60.4C4 D15⋊2C8 D30.5C4 C3×C4.Dic5 C10×C3⋊C8 D60⋊11C2 Dic30 D60 C15⋊7D4 C4.Dic5 C2×C3⋊C8 C5⋊2C8 C2×C20 C3⋊C8 C2×C12 C20 C2×C10 C15 C12 C2×C6 C3 C2×C4 C5 C4 C4 C22 C1 # reps 1 2 2 1 1 1 2 2 4 1 2 2 1 4 2 2 2 4 4 4 16 2 2 2 2 2 8

Matrix representation of D60.4C4 in GL4(𝔽241) generated by

 197 78 0 0 163 78 0 0 0 0 0 1 0 0 240 1
,
 240 51 0 0 0 1 0 0 0 0 1 240 0 0 0 240
,
 115 90 0 0 171 126 0 0 0 0 64 0 0 0 0 64
`G:=sub<GL(4,GF(241))| [197,163,0,0,78,78,0,0,0,0,0,240,0,0,1,1],[240,0,0,0,51,1,0,0,0,0,1,0,0,0,240,240],[115,171,0,0,90,126,0,0,0,0,64,0,0,0,0,64] >;`

D60.4C4 in GAP, Magma, Sage, TeX

`D_{60}._4C_4`
`% in TeX`

`G:=Group("D60.4C4");`
`// GroupNames label`

`G:=SmallGroup(480,367);`
`// by ID`

`G=gap.SmallGroup(480,367);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,64,422,80,1356,18822]);`
`// 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^19,c*b*c^-1=a^18*b>;`
`// generators/relations`

׿
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