Copied to
clipboard

## G = C30.12D6order 360 = 23·32·5

### 12nd non-split extension by C30 of D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C30 — C30.12D6
 Chief series C1 — C5 — C15 — C3×C15 — C3×C30 — D5×C3×C6 — C30.12D6
 Lower central C3×C15 — C3×C30 — C30.12D6
 Upper central C1 — C2

Generators and relations for C30.12D6
G = < a,b,c | a30=b6=1, c2=a15, bab-1=a19, cac-1=a-1, cbc-1=a15b-1 >

Subgroups: 528 in 96 conjugacy classes, 34 normal (14 characteristic)
C1, C2, C2, C3, C4, C22, C5, S3, C6, C6, D4, C32, D5, C10, C10, Dic3, D6, C2×C6, C15, C3⋊S3, C3×C6, C3×C6, Dic5, D10, C2×C10, C3⋊D4, C5×S3, C3×D5, C30, C3⋊Dic3, C2×C3⋊S3, C62, C5⋊D4, C3×C15, Dic15, C6×D5, S3×C10, C327D4, C32×D5, C5×C3⋊S3, C3×C30, C15⋊D4, C3⋊Dic15, D5×C3×C6, C10×C3⋊S3, C30.12D6
Quotients: C1, C2, C22, S3, D4, D5, D6, C3⋊S3, D10, C3⋊D4, C2×C3⋊S3, C5⋊D4, S3×D5, C327D4, C15⋊D4, D5×C3⋊S3, C30.12D6

Smallest permutation representation of C30.12D6
On 180 points
Generators in S180
```(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)
(1 131 116 173 72 57)(2 150 117 162 73 46)(3 139 118 151 74 35)(4 128 119 170 75 54)(5 147 120 159 76 43)(6 136 91 178 77 32)(7 125 92 167 78 51)(8 144 93 156 79 40)(9 133 94 175 80 59)(10 122 95 164 81 48)(11 141 96 153 82 37)(12 130 97 172 83 56)(13 149 98 161 84 45)(14 138 99 180 85 34)(15 127 100 169 86 53)(16 146 101 158 87 42)(17 135 102 177 88 31)(18 124 103 166 89 50)(19 143 104 155 90 39)(20 132 105 174 61 58)(21 121 106 163 62 47)(22 140 107 152 63 36)(23 129 108 171 64 55)(24 148 109 160 65 44)(25 137 110 179 66 33)(26 126 111 168 67 52)(27 145 112 157 68 41)(28 134 113 176 69 60)(29 123 114 165 70 49)(30 142 115 154 71 38)
(1 57 16 42)(2 56 17 41)(3 55 18 40)(4 54 19 39)(5 53 20 38)(6 52 21 37)(7 51 22 36)(8 50 23 35)(9 49 24 34)(10 48 25 33)(11 47 26 32)(12 46 27 31)(13 45 28 60)(14 44 29 59)(15 43 30 58)(61 142 76 127)(62 141 77 126)(63 140 78 125)(64 139 79 124)(65 138 80 123)(66 137 81 122)(67 136 82 121)(68 135 83 150)(69 134 84 149)(70 133 85 148)(71 132 86 147)(72 131 87 146)(73 130 88 145)(74 129 89 144)(75 128 90 143)(91 168 106 153)(92 167 107 152)(93 166 108 151)(94 165 109 180)(95 164 110 179)(96 163 111 178)(97 162 112 177)(98 161 113 176)(99 160 114 175)(100 159 115 174)(101 158 116 173)(102 157 117 172)(103 156 118 171)(104 155 119 170)(105 154 120 169)```

`G:=sub<Sym(180)| (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), (1,131,116,173,72,57)(2,150,117,162,73,46)(3,139,118,151,74,35)(4,128,119,170,75,54)(5,147,120,159,76,43)(6,136,91,178,77,32)(7,125,92,167,78,51)(8,144,93,156,79,40)(9,133,94,175,80,59)(10,122,95,164,81,48)(11,141,96,153,82,37)(12,130,97,172,83,56)(13,149,98,161,84,45)(14,138,99,180,85,34)(15,127,100,169,86,53)(16,146,101,158,87,42)(17,135,102,177,88,31)(18,124,103,166,89,50)(19,143,104,155,90,39)(20,132,105,174,61,58)(21,121,106,163,62,47)(22,140,107,152,63,36)(23,129,108,171,64,55)(24,148,109,160,65,44)(25,137,110,179,66,33)(26,126,111,168,67,52)(27,145,112,157,68,41)(28,134,113,176,69,60)(29,123,114,165,70,49)(30,142,115,154,71,38), (1,57,16,42)(2,56,17,41)(3,55,18,40)(4,54,19,39)(5,53,20,38)(6,52,21,37)(7,51,22,36)(8,50,23,35)(9,49,24,34)(10,48,25,33)(11,47,26,32)(12,46,27,31)(13,45,28,60)(14,44,29,59)(15,43,30,58)(61,142,76,127)(62,141,77,126)(63,140,78,125)(64,139,79,124)(65,138,80,123)(66,137,81,122)(67,136,82,121)(68,135,83,150)(69,134,84,149)(70,133,85,148)(71,132,86,147)(72,131,87,146)(73,130,88,145)(74,129,89,144)(75,128,90,143)(91,168,106,153)(92,167,107,152)(93,166,108,151)(94,165,109,180)(95,164,110,179)(96,163,111,178)(97,162,112,177)(98,161,113,176)(99,160,114,175)(100,159,115,174)(101,158,116,173)(102,157,117,172)(103,156,118,171)(104,155,119,170)(105,154,120,169)>;`

`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), (1,131,116,173,72,57)(2,150,117,162,73,46)(3,139,118,151,74,35)(4,128,119,170,75,54)(5,147,120,159,76,43)(6,136,91,178,77,32)(7,125,92,167,78,51)(8,144,93,156,79,40)(9,133,94,175,80,59)(10,122,95,164,81,48)(11,141,96,153,82,37)(12,130,97,172,83,56)(13,149,98,161,84,45)(14,138,99,180,85,34)(15,127,100,169,86,53)(16,146,101,158,87,42)(17,135,102,177,88,31)(18,124,103,166,89,50)(19,143,104,155,90,39)(20,132,105,174,61,58)(21,121,106,163,62,47)(22,140,107,152,63,36)(23,129,108,171,64,55)(24,148,109,160,65,44)(25,137,110,179,66,33)(26,126,111,168,67,52)(27,145,112,157,68,41)(28,134,113,176,69,60)(29,123,114,165,70,49)(30,142,115,154,71,38), (1,57,16,42)(2,56,17,41)(3,55,18,40)(4,54,19,39)(5,53,20,38)(6,52,21,37)(7,51,22,36)(8,50,23,35)(9,49,24,34)(10,48,25,33)(11,47,26,32)(12,46,27,31)(13,45,28,60)(14,44,29,59)(15,43,30,58)(61,142,76,127)(62,141,77,126)(63,140,78,125)(64,139,79,124)(65,138,80,123)(66,137,81,122)(67,136,82,121)(68,135,83,150)(69,134,84,149)(70,133,85,148)(71,132,86,147)(72,131,87,146)(73,130,88,145)(74,129,89,144)(75,128,90,143)(91,168,106,153)(92,167,107,152)(93,166,108,151)(94,165,109,180)(95,164,110,179)(96,163,111,178)(97,162,112,177)(98,161,113,176)(99,160,114,175)(100,159,115,174)(101,158,116,173)(102,157,117,172)(103,156,118,171)(104,155,119,170)(105,154,120,169) );`

`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)], [(1,131,116,173,72,57),(2,150,117,162,73,46),(3,139,118,151,74,35),(4,128,119,170,75,54),(5,147,120,159,76,43),(6,136,91,178,77,32),(7,125,92,167,78,51),(8,144,93,156,79,40),(9,133,94,175,80,59),(10,122,95,164,81,48),(11,141,96,153,82,37),(12,130,97,172,83,56),(13,149,98,161,84,45),(14,138,99,180,85,34),(15,127,100,169,86,53),(16,146,101,158,87,42),(17,135,102,177,88,31),(18,124,103,166,89,50),(19,143,104,155,90,39),(20,132,105,174,61,58),(21,121,106,163,62,47),(22,140,107,152,63,36),(23,129,108,171,64,55),(24,148,109,160,65,44),(25,137,110,179,66,33),(26,126,111,168,67,52),(27,145,112,157,68,41),(28,134,113,176,69,60),(29,123,114,165,70,49),(30,142,115,154,71,38)], [(1,57,16,42),(2,56,17,41),(3,55,18,40),(4,54,19,39),(5,53,20,38),(6,52,21,37),(7,51,22,36),(8,50,23,35),(9,49,24,34),(10,48,25,33),(11,47,26,32),(12,46,27,31),(13,45,28,60),(14,44,29,59),(15,43,30,58),(61,142,76,127),(62,141,77,126),(63,140,78,125),(64,139,79,124),(65,138,80,123),(66,137,81,122),(67,136,82,121),(68,135,83,150),(69,134,84,149),(70,133,85,148),(71,132,86,147),(72,131,87,146),(73,130,88,145),(74,129,89,144),(75,128,90,143),(91,168,106,153),(92,167,107,152),(93,166,108,151),(94,165,109,180),(95,164,110,179),(96,163,111,178),(97,162,112,177),(98,161,113,176),(99,160,114,175),(100,159,115,174),(101,158,116,173),(102,157,117,172),(103,156,118,171),(104,155,119,170),(105,154,120,169)]])`

45 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 4 5A 5B 6A 6B 6C 6D 6E ··· 6L 10A 10B 10C 10D 10E 10F 15A ··· 15H 30A ··· 30H order 1 2 2 2 3 3 3 3 4 5 5 6 6 6 6 6 ··· 6 10 10 10 10 10 10 15 ··· 15 30 ··· 30 size 1 1 10 18 2 2 2 2 90 2 2 2 2 2 2 10 ··· 10 2 2 18 18 18 18 4 ··· 4 4 ··· 4

45 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + - image C1 C2 C2 C2 S3 D4 D5 D6 D10 C3⋊D4 C5⋊D4 S3×D5 C15⋊D4 kernel C30.12D6 C3⋊Dic15 D5×C3×C6 C10×C3⋊S3 C6×D5 C3×C15 C2×C3⋊S3 C30 C3×C6 C15 C32 C6 C3 # reps 1 1 1 1 4 1 2 4 2 8 4 8 8

Matrix representation of C30.12D6 in GL6(𝔽61)

 41 40 0 0 0 0 0 3 0 0 0 0 0 0 60 0 0 0 0 0 0 60 0 0 0 0 0 0 1 1 0 0 0 0 60 0
,
 1 0 0 0 0 0 57 60 0 0 0 0 0 0 52 9 0 0 0 0 52 43 0 0 0 0 0 0 9 52 0 0 0 0 9 18
,
 1 31 0 0 0 0 57 60 0 0 0 0 0 0 9 52 0 0 0 0 43 52 0 0 0 0 0 0 43 52 0 0 0 0 9 18

`G:=sub<GL(6,GF(61))| [41,0,0,0,0,0,40,3,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,60,0,0,0,0,1,0],[1,57,0,0,0,0,0,60,0,0,0,0,0,0,52,52,0,0,0,0,9,43,0,0,0,0,0,0,9,9,0,0,0,0,52,18],[1,57,0,0,0,0,31,60,0,0,0,0,0,0,9,43,0,0,0,0,52,52,0,0,0,0,0,0,43,9,0,0,0,0,52,18] >;`

C30.12D6 in GAP, Magma, Sage, TeX

`C_{30}._{12}D_6`
`% in TeX`

`G:=Group("C30.12D6");`
`// GroupNames label`

`G:=SmallGroup(360,68);`
`// by ID`

`G=gap.SmallGroup(360,68);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-3,-3,-5,73,201,730,10373]);`
`// Polycyclic`

`G:=Group<a,b,c|a^30=b^6=1,c^2=a^15,b*a*b^-1=a^19,c*a*c^-1=a^-1,c*b*c^-1=a^15*b^-1>;`
`// generators/relations`

׿
×
𝔽