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## G = C60.68D4order 480 = 25·3·5

### 68th non-split extension by C60 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C30 — C60.68D4
 Chief series C1 — C5 — C15 — C30 — C2×C30 — D5×C2×C6 — D10⋊Dic3 — C60.68D4
 Lower central C15 — C2×C30 — C60.68D4
 Upper central C1 — C22 — C2×C4

Generators and relations for C60.68D4
G = < a,b,c | a60=b4=1, c2=a30, bab-1=a11, cac-1=a-1, cbc-1=a30b-1 >

Subgroups: 700 in 148 conjugacy classes, 56 normal (34 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, C6, C6, C2×C4, C2×C4, Q8, C23, D5, C10, Dic3, C12, C12, C2×C6, C2×C6, C15, C22⋊C4, C4⋊C4, C22×C4, C2×Q8, Dic5, C20, C20, D10, D10, C2×C10, Dic6, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×C6, C3×D5, C30, C22⋊Q8, Dic10, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, Dic3⋊C4, C4⋊Dic3, C6.D4, C2×Dic6, C22×C12, C5×Dic3, C3×Dic5, Dic15, C60, C6×D5, C6×D5, C2×C30, C4⋊Dic5, D10⋊C4, C5×C4⋊C4, C2×Dic10, C2×C4×D5, C12.48D4, D5×C12, C6×Dic5, C10×Dic3, Dic30, C2×Dic15, C2×C60, D5×C2×C6, D102Q8, D10⋊Dic3, C6.Dic10, C5×C4⋊Dic3, D5×C2×C12, C2×Dic30, C60.68D4
Quotients: C1, C2, C22, S3, D4, Q8, C23, D5, D6, C2×D4, C2×Q8, C4○D4, D10, Dic6, C3⋊D4, C22×S3, C22⋊Q8, D20, C22×D5, C2×Dic6, C4○D12, C2×C3⋊D4, S3×D5, C2×D20, D42D5, Q8×D5, C12.48D4, C3⋊D20, C2×S3×D5, D102Q8, D5×Dic6, D125D5, C2×C3⋊D20, C60.68D4

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

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

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

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

66 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 12C 12D 12E 12F 12G 12H 15A 15B 20A 20B 20C 20D 20E ··· 20L 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 12 12 12 12 12 12 15 15 20 20 20 20 20 ··· 20 30 ··· 30 60 ··· 60 size 1 1 1 1 10 10 2 2 2 10 10 12 12 60 60 2 2 2 2 2 10 10 10 10 2 ··· 2 2 2 2 2 10 10 10 10 4 4 4 4 4 4 12 ··· 12 4 ··· 4 4 ··· 4

66 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 type + + + + + + + + - + + + + + + - + + - - + + - - image C1 C2 C2 C2 C2 C2 S3 D4 Q8 D5 D6 D6 D6 C4○D4 D10 D10 C3⋊D4 Dic6 D20 C4○D12 S3×D5 D4⋊2D5 Q8×D5 C3⋊D20 C2×S3×D5 D5×Dic6 D12⋊5D5 kernel C60.68D4 D10⋊Dic3 C6.Dic10 C5×C4⋊Dic3 D5×C2×C12 C2×Dic30 C2×C4×D5 C60 C6×D5 C4⋊Dic3 C2×Dic5 C2×C20 C22×D5 C30 C2×Dic3 C2×C12 C20 D10 C12 C10 C2×C4 C6 C6 C4 C22 C2 C2 # reps 1 2 2 1 1 1 1 2 2 2 1 1 1 2 4 2 4 4 8 4 2 2 2 4 2 4 4

Matrix representation of C60.68D4 in GL4(𝔽61) generated by

 21 0 0 0 44 32 0 0 0 0 1 17 0 0 44 17
,
 11 18 0 0 34 50 0 0 0 0 29 7 0 0 54 32
,
 11 18 0 0 0 50 0 0 0 0 2 2 0 0 29 59
`G:=sub<GL(4,GF(61))| [21,44,0,0,0,32,0,0,0,0,1,44,0,0,17,17],[11,34,0,0,18,50,0,0,0,0,29,54,0,0,7,32],[11,0,0,0,18,50,0,0,0,0,2,29,0,0,2,59] >;`

C60.68D4 in GAP, Magma, Sage, TeX

`C_{60}._{68}D_4`
`% in TeX`

`G:=Group("C60.68D4");`
`// GroupNames label`

`G:=SmallGroup(480,436);`
`// by ID`

`G=gap.SmallGroup(480,436);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,141,176,219,100,1356,18822]);`
`// Polycyclic`

`G:=Group<a,b,c|a^60=b^4=1,c^2=a^30,b*a*b^-1=a^11,c*a*c^-1=a^-1,c*b*c^-1=a^30*b^-1>;`
`// generators/relations`

׿
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