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G = C60.93D4order 480 = 25·3·5

93rd non-split extension by C60 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — C60.93D4
 Chief series C1 — C5 — C15 — C30 — C60 — C2×C60 — D5×C2×C12 — C60.93D4
 Lower central C15 — C30 — C60.93D4
 Upper central C1 — C2×C4

Generators and relations for C60.93D4
G = < a,b,c | a60=c2=1, b4=a30, bab-1=a29, cac=a49, cbc=a15b3 >

Subgroups: 380 in 100 conjugacy classes, 46 normal (42 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×2], C4, C22, C22 [×4], C5, C6 [×3], C6 [×2], C8 [×2], C2×C4, C2×C4 [×3], C23, D5 [×2], C10 [×3], C12 [×2], C12, C2×C6, C2×C6 [×4], C15, C2×C8 [×2], C22×C4, Dic5, C20 [×2], D10 [×2], D10 [×2], C2×C10, C3⋊C8 [×2], C2×C12, C2×C12 [×3], C22×C6, C3×D5 [×2], C30 [×3], C22⋊C8, C52C8, C40, C4×D5 [×2], C2×Dic5, C2×C20, C22×D5, C2×C3⋊C8, C2×C3⋊C8, C22×C12, C3×Dic5, C60 [×2], C6×D5 [×2], C6×D5 [×2], C2×C30, C2×C52C8, C2×C40, C2×C4×D5, C12.55D4, C5×C3⋊C8, C153C8, D5×C12 [×2], C6×Dic5, C2×C60, D5×C2×C6, D101C8, C10×C3⋊C8, C2×C153C8, D5×C2×C12, C60.93D4
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C8 [×2], C2×C4, D4 [×2], D5, Dic3 [×2], D6, C22⋊C4, C2×C8, M4(2), D10, C3⋊C8 [×2], C2×Dic3, C3⋊D4 [×2], C22⋊C8, C4×D5, D20, C5⋊D4, C2×C3⋊C8, C4.Dic3, C6.D4, S3×D5, C8×D5, C8⋊D5, D10⋊C4, C12.55D4, D5×Dic3, C15⋊D4, C3⋊D20, D101C8, D5×C3⋊C8, C20.32D6, D10⋊Dic3, C60.93D4

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

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

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

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

84 conjugacy classes

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

84 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + + - + - + + + - + - image C1 C2 C2 C2 C4 C4 C8 S3 D4 D5 Dic3 D6 Dic3 M4(2) D10 C3⋊D4 C3⋊C8 D20 C5⋊D4 C4×D5 C4.Dic3 C8×D5 C8⋊D5 S3×D5 C15⋊D4 C3⋊D20 D5×Dic3 D5×C3⋊C8 C20.32D6 kernel C60.93D4 C10×C3⋊C8 C2×C15⋊3C8 D5×C2×C12 C6×Dic5 D5×C2×C6 C6×D5 C2×C4×D5 C60 C2×C3⋊C8 C2×Dic5 C2×C20 C22×D5 C30 C2×C12 C20 D10 C12 C12 C2×C6 C10 C6 C6 C2×C4 C4 C4 C22 C2 C2 # reps 1 1 1 1 2 2 8 1 2 2 1 1 1 2 2 4 4 4 4 4 4 8 8 2 2 2 2 4 4

Matrix representation of C60.93D4 in GL4(𝔽241) generated by

 189 189 0 0 52 1 0 0 0 0 4 0 0 0 81 60
,
 200 156 0 0 119 41 0 0 0 0 214 142 0 0 64 27
,
 1 0 0 0 189 240 0 0 0 0 240 0 0 0 0 240
`G:=sub<GL(4,GF(241))| [189,52,0,0,189,1,0,0,0,0,4,81,0,0,0,60],[200,119,0,0,156,41,0,0,0,0,214,64,0,0,142,27],[1,189,0,0,0,240,0,0,0,0,240,0,0,0,0,240] >;`

C60.93D4 in GAP, Magma, Sage, TeX

`C_{60}._{93}D_4`
`% in TeX`

`G:=Group("C60.93D4");`
`// GroupNames label`

`G:=SmallGroup(480,31);`
`// by ID`

`G=gap.SmallGroup(480,31);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,141,36,100,1356,18822]);`
`// Polycyclic`

`G:=Group<a,b,c|a^60=c^2=1,b^4=a^30,b*a*b^-1=a^29,c*a*c=a^49,c*b*c=a^15*b^3>;`
`// generators/relations`

׿
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