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## G = D60.5C4order 480 = 25·3·5

### 3rd non-split extension by D60 of C4 acting via C4/C2=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — D60.5C4
 Chief series C1 — C5 — C15 — C30 — C60 — C3×C5⋊2C8 — D15⋊2C8 — D60.5C4
 Lower central C15 — C30 — D60.5C4
 Upper central C1 — C4 — C2×C4

Generators and relations for D60.5C4
G = < a,b,c | a60=b2=1, c4=a30, bab=a-1, cac-1=a49, cbc-1=a48b >

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], C4.Dic3, C2×C24, C4○D12, Dic15 [×2], C60 [×2], D30 [×2], C2×C30, C8×D5 [×2], C8⋊D5 [×2], C2×C52C8, C5×M4(2), C4○D20, C8○D12, C5×C3⋊C8 [×2], C3×C52C8 [×2], Dic30, C4×D15 [×2], D60, C157D4 [×2], C2×C60, D20.2C4, D152C8 [×2], D30.5C4 [×2], C6×C52C8, C5×C4.Dic3, D6011C2, D60.5C4
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, C8○D12, D30.C2 [×2], C2×S3×D5, D20.2C4, C2×D30.C2, D60.5C4

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

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

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

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

72 conjugacy classes

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

72 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 C8○D12 S3×D5 D30.C2 C2×S3×D5 D30.C2 D20.2C4 D60.5C4 kernel D60.5C4 D15⋊2C8 D30.5C4 C6×C5⋊2C8 C5×C4.Dic3 D60⋊11C2 Dic30 D60 C15⋊7D4 C2×C5⋊2C8 C4.Dic3 C5⋊2C8 C2×C20 C3⋊C8 C2×C12 C20 C2×C10 C15 C12 C2×C6 C5 C2×C4 C4 C4 C22 C3 C1 # reps 1 2 2 1 1 1 2 2 4 1 2 2 1 4 2 2 2 4 4 4 8 2 2 2 2 4 8

Matrix representation of D60.5C4 in GL4(𝔽241) generated by

 0 52 0 0 190 190 0 0 0 0 198 142 0 0 99 99
,
 1 0 0 0 50 240 0 0 0 0 1 240 0 0 0 240
,
 51 52 0 0 191 190 0 0 0 0 30 0 0 0 0 30
`G:=sub<GL(4,GF(241))| [0,190,0,0,52,190,0,0,0,0,198,99,0,0,142,99],[1,50,0,0,0,240,0,0,0,0,1,0,0,0,240,240],[51,191,0,0,52,190,0,0,0,0,30,0,0,0,0,30] >;`

D60.5C4 in GAP, Magma, Sage, TeX

`D_{60}._5C_4`
`% in TeX`

`G:=Group("D60.5C4");`
`// GroupNames label`

`G:=SmallGroup(480,366);`
`// by ID`

`G=gap.SmallGroup(480,366);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,64,219,80,1356,18822]);`
`// 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^49,c*b*c^-1=a^48*b>;`
`// generators/relations`

׿
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