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## G = C30.D6order 360 = 23·32·5

### 11st non-split extension by C30 of D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C15 — C30.D6
 Chief series C1 — C5 — C15 — C3×C15 — C3×C30 — C32×Dic5 — C30.D6
 Lower central C3×C15 — C30.D6
 Upper central C1 — C2

Generators and relations for C30.D6
G = < a,b,c | a30=c2=1, b6=a15, bab-1=a19, cac=a-1, cbc=b5 >

Subgroups: 632 in 96 conjugacy classes, 36 normal (14 characteristic)
C1, C2, C2, C3, C4, C22, C5, S3, C6, C2×C4, C32, D5, C10, Dic3, C12, D6, C15, C3⋊S3, C3×C6, Dic5, C20, D10, C4×S3, D15, C30, C3⋊Dic3, C3×C12, C2×C3⋊S3, C4×D5, C3×C15, C5×Dic3, C3×Dic5, D30, C4×C3⋊S3, C3⋊D15, C3×C30, D30.C2, C32×Dic5, C5×C3⋊Dic3, C2×C3⋊D15, C30.D6
Quotients: C1, C2, C4, C22, S3, C2×C4, D5, D6, C3⋊S3, D10, C4×S3, C2×C3⋊S3, C4×D5, S3×D5, C4×C3⋊S3, D30.C2, D5×C3⋊S3, C30.D6

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

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

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

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

48 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 4A 4B 4C 4D 5A 5B 6A 6B 6C 6D 10A 10B 12A ··· 12H 15A ··· 15H 20A 20B 20C 20D 30A ··· 30H order 1 2 2 2 3 3 3 3 4 4 4 4 5 5 6 6 6 6 10 10 12 ··· 12 15 ··· 15 20 20 20 20 30 ··· 30 size 1 1 45 45 2 2 2 2 5 5 9 9 2 2 2 2 2 2 2 2 10 ··· 10 4 ··· 4 18 18 18 18 4 ··· 4

48 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + + + + + image C1 C2 C2 C2 C4 S3 D5 D6 D10 C4×S3 C4×D5 S3×D5 D30.C2 kernel C30.D6 C32×Dic5 C5×C3⋊Dic3 C2×C3⋊D15 C3⋊D15 C3×Dic5 C3⋊Dic3 C30 C3×C6 C15 C32 C6 C3 # reps 1 1 1 1 4 4 2 4 2 8 4 8 8

Matrix representation of C30.D6 in GL6(𝔽61)

 60 60 0 0 0 0 1 0 0 0 0 0 0 0 60 1 0 0 0 0 60 0 0 0 0 0 0 0 1 60 0 0 0 0 45 17
,
 0 1 0 0 0 0 60 60 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 4 11 0 0 0 0 4 57
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 17 1 0 0 0 0 17 44

`G:=sub<GL(6,GF(61))| [60,1,0,0,0,0,60,0,0,0,0,0,0,0,60,60,0,0,0,0,1,0,0,0,0,0,0,0,1,45,0,0,0,0,60,17],[0,60,0,0,0,0,1,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,4,0,0,0,0,11,57],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,17,17,0,0,0,0,1,44] >;`

C30.D6 in GAP, Magma, Sage, TeX

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

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

`G:=SmallGroup(360,67);`
`// by ID`

`G=gap.SmallGroup(360,67);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-3,-3,-5,24,31,201,730,10373]);`
`// Polycyclic`

`G:=Group<a,b,c|a^30=c^2=1,b^6=a^15,b*a*b^-1=a^19,c*a*c=a^-1,c*b*c=b^5>;`
`// generators/relations`

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