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## G = C62.30D6order 432 = 24·33

### 13rd non-split extension by C62 of D6 acting via D6/C2=S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C2×He3 — C62.30D6
 Chief series C1 — C3 — C32 — He3 — C2×He3 — C22×He3 — C2×He3⋊3C4 — C62.30D6
 Lower central He3 — C2×He3 — C62.30D6
 Upper central C1 — C2×C6 — C2×C12

Generators and relations for C62.30D6
G = < a,b,c,d | a6=b6=1, c6=b3, d2=a3, ab=ba, cac-1=ab2, dad-1=a-1b4, bc=cb, bd=db, dcd-1=b3c5 >

Subgroups: 453 in 143 conjugacy classes, 59 normal (19 characteristic)
C1, C2 [×3], C3, C3 [×4], C4 [×2], C4 [×2], C22, C6 [×3], C6 [×12], C2×C4, C2×C4 [×2], C32 [×4], Dic3 [×8], C12 [×2], C12 [×10], C2×C6, C2×C6 [×4], C4⋊C4, C3×C6 [×12], C2×Dic3 [×8], C2×C12, C2×C12 [×6], He3, C3×Dic3 [×8], C3×C12 [×8], C62 [×4], C4⋊Dic3 [×4], C3×C4⋊C4, C2×He3 [×3], C6×Dic3 [×8], C6×C12 [×4], He33C4 [×2], C4×He3 [×2], C22×He3, C3×C4⋊Dic3 [×4], C2×He33C4 [×2], C2×C4×He3, C62.30D6
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×4], C2×C4, D4, Q8, Dic3 [×8], D6 [×4], C4⋊C4, C3⋊S3, Dic6 [×4], D12 [×4], C2×Dic3 [×4], C3⋊Dic3 [×2], C2×C3⋊S3, C4⋊Dic3 [×4], He3⋊C2, C324Q8, C12⋊S3, C2×C3⋊Dic3, He33C4 [×2], C2×He3⋊C2, C12⋊Dic3, He34Q8, He35D4, C2×He33C4, C62.30D6

Smallest permutation representation of C62.30D6
On 144 points
Generators in S144
```(1 75 67 36 112 121)(2 84 64 33 109 130)(3 81 61 34 118 127)(4 78 70 35 115 124)(5 105 86 28 133 49)(6 102 95 25 142 58)(7 99 92 26 139 55)(8 108 89 27 136 52)(9 74 62 30 111 128)(10 83 71 31 120 125)(11 80 68 32 117 122)(12 77 65 29 114 131)(13 141 53 42 101 90)(14 138 50 43 98 87)(15 135 59 44 107 96)(16 144 56 41 104 93)(17 137 57 46 97 94)(18 134 54 47 106 91)(19 143 51 48 103 88)(20 140 60 45 100 85)(21 110 123 37 73 69)(22 119 132 38 82 66)(23 116 129 39 79 63)(24 113 126 40 76 72)
(1 37 10 3 39 12)(2 38 11 4 40 9)(5 48 42 7 46 44)(6 45 43 8 47 41)(13 26 17 15 28 19)(14 27 18 16 25 20)(21 31 34 23 29 36)(22 32 35 24 30 33)(49 51 53 55 57 59)(50 52 54 56 58 60)(61 63 65 67 69 71)(62 64 66 68 70 72)(73 83 81 79 77 75)(74 84 82 80 78 76)(85 87 89 91 93 95)(86 88 90 92 94 96)(97 107 105 103 101 99)(98 108 106 104 102 100)(109 119 117 115 113 111)(110 120 118 116 114 112)(121 123 125 127 129 131)(122 124 126 128 130 132)(133 143 141 139 137 135)(134 144 142 140 138 136)
(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)
(1 43 36 14)(2 42 33 13)(3 41 34 16)(4 44 35 15)(5 24 28 40)(6 23 25 39)(7 22 26 38)(8 21 27 37)(9 48 30 19)(10 47 31 18)(11 46 32 17)(12 45 29 20)(49 109 86 84)(50 120 87 83)(51 119 88 82)(52 118 89 81)(53 117 90 80)(54 116 91 79)(55 115 92 78)(56 114 93 77)(57 113 94 76)(58 112 95 75)(59 111 96 74)(60 110 85 73)(61 140 127 100)(62 139 128 99)(63 138 129 98)(64 137 130 97)(65 136 131 108)(66 135 132 107)(67 134 121 106)(68 133 122 105)(69 144 123 104)(70 143 124 103)(71 142 125 102)(72 141 126 101)```

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

`G:=Group( (1,75,67,36,112,121)(2,84,64,33,109,130)(3,81,61,34,118,127)(4,78,70,35,115,124)(5,105,86,28,133,49)(6,102,95,25,142,58)(7,99,92,26,139,55)(8,108,89,27,136,52)(9,74,62,30,111,128)(10,83,71,31,120,125)(11,80,68,32,117,122)(12,77,65,29,114,131)(13,141,53,42,101,90)(14,138,50,43,98,87)(15,135,59,44,107,96)(16,144,56,41,104,93)(17,137,57,46,97,94)(18,134,54,47,106,91)(19,143,51,48,103,88)(20,140,60,45,100,85)(21,110,123,37,73,69)(22,119,132,38,82,66)(23,116,129,39,79,63)(24,113,126,40,76,72), (1,37,10,3,39,12)(2,38,11,4,40,9)(5,48,42,7,46,44)(6,45,43,8,47,41)(13,26,17,15,28,19)(14,27,18,16,25,20)(21,31,34,23,29,36)(22,32,35,24,30,33)(49,51,53,55,57,59)(50,52,54,56,58,60)(61,63,65,67,69,71)(62,64,66,68,70,72)(73,83,81,79,77,75)(74,84,82,80,78,76)(85,87,89,91,93,95)(86,88,90,92,94,96)(97,107,105,103,101,99)(98,108,106,104,102,100)(109,119,117,115,113,111)(110,120,118,116,114,112)(121,123,125,127,129,131)(122,124,126,128,130,132)(133,143,141,139,137,135)(134,144,142,140,138,136), (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), (1,43,36,14)(2,42,33,13)(3,41,34,16)(4,44,35,15)(5,24,28,40)(6,23,25,39)(7,22,26,38)(8,21,27,37)(9,48,30,19)(10,47,31,18)(11,46,32,17)(12,45,29,20)(49,109,86,84)(50,120,87,83)(51,119,88,82)(52,118,89,81)(53,117,90,80)(54,116,91,79)(55,115,92,78)(56,114,93,77)(57,113,94,76)(58,112,95,75)(59,111,96,74)(60,110,85,73)(61,140,127,100)(62,139,128,99)(63,138,129,98)(64,137,130,97)(65,136,131,108)(66,135,132,107)(67,134,121,106)(68,133,122,105)(69,144,123,104)(70,143,124,103)(71,142,125,102)(72,141,126,101) );`

`G=PermutationGroup([(1,75,67,36,112,121),(2,84,64,33,109,130),(3,81,61,34,118,127),(4,78,70,35,115,124),(5,105,86,28,133,49),(6,102,95,25,142,58),(7,99,92,26,139,55),(8,108,89,27,136,52),(9,74,62,30,111,128),(10,83,71,31,120,125),(11,80,68,32,117,122),(12,77,65,29,114,131),(13,141,53,42,101,90),(14,138,50,43,98,87),(15,135,59,44,107,96),(16,144,56,41,104,93),(17,137,57,46,97,94),(18,134,54,47,106,91),(19,143,51,48,103,88),(20,140,60,45,100,85),(21,110,123,37,73,69),(22,119,132,38,82,66),(23,116,129,39,79,63),(24,113,126,40,76,72)], [(1,37,10,3,39,12),(2,38,11,4,40,9),(5,48,42,7,46,44),(6,45,43,8,47,41),(13,26,17,15,28,19),(14,27,18,16,25,20),(21,31,34,23,29,36),(22,32,35,24,30,33),(49,51,53,55,57,59),(50,52,54,56,58,60),(61,63,65,67,69,71),(62,64,66,68,70,72),(73,83,81,79,77,75),(74,84,82,80,78,76),(85,87,89,91,93,95),(86,88,90,92,94,96),(97,107,105,103,101,99),(98,108,106,104,102,100),(109,119,117,115,113,111),(110,120,118,116,114,112),(121,123,125,127,129,131),(122,124,126,128,130,132),(133,143,141,139,137,135),(134,144,142,140,138,136)], [(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)], [(1,43,36,14),(2,42,33,13),(3,41,34,16),(4,44,35,15),(5,24,28,40),(6,23,25,39),(7,22,26,38),(8,21,27,37),(9,48,30,19),(10,47,31,18),(11,46,32,17),(12,45,29,20),(49,109,86,84),(50,120,87,83),(51,119,88,82),(52,118,89,81),(53,117,90,80),(54,116,91,79),(55,115,92,78),(56,114,93,77),(57,113,94,76),(58,112,95,75),(59,111,96,74),(60,110,85,73),(61,140,127,100),(62,139,128,99),(63,138,129,98),(64,137,130,97),(65,136,131,108),(66,135,132,107),(67,134,121,106),(68,133,122,105),(69,144,123,104),(70,143,124,103),(71,142,125,102),(72,141,126,101)])`

62 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 3F 4A 4B 4C 4D 4E 4F 6A ··· 6F 6G ··· 6R 12A 12B 12C 12D 12E ··· 12T 12U ··· 12AB order 1 2 2 2 3 3 3 3 3 3 4 4 4 4 4 4 6 ··· 6 6 ··· 6 12 12 12 12 12 ··· 12 12 ··· 12 size 1 1 1 1 1 1 6 6 6 6 2 2 18 18 18 18 1 ··· 1 6 ··· 6 2 2 2 2 6 ··· 6 18 ··· 18

62 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 3 3 3 6 6 type + + + + + - - + - + image C1 C2 C2 C4 S3 D4 Q8 Dic3 D6 Dic6 D12 He3⋊C2 He3⋊3C4 C2×He3⋊C2 He3⋊4Q8 He3⋊5D4 kernel C62.30D6 C2×He3⋊3C4 C2×C4×He3 C4×He3 C6×C12 C2×He3 C2×He3 C3×C12 C62 C3×C6 C3×C6 C2×C4 C4 C22 C2 C2 # reps 1 2 1 4 4 1 1 8 4 8 8 4 8 4 2 2

Matrix representation of C62.30D6 in GL7(𝔽13)

 0 1 0 0 0 0 0 12 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0
,
 12 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 9
,
 6 3 0 0 0 0 0 10 3 0 0 0 0 0 0 0 3 10 0 0 0 0 0 3 6 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 9
,
 7 10 0 0 0 0 0 3 6 0 0 0 0 0 0 0 2 11 0 0 0 0 0 9 11 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 4 0 0 0 0 0 10 0

`G:=sub<GL(7,GF(13))| [0,12,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,12,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0],[12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,9],[6,10,0,0,0,0,0,3,3,0,0,0,0,0,0,0,3,3,0,0,0,0,0,10,6,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,9],[7,3,0,0,0,0,0,10,6,0,0,0,0,0,0,0,2,9,0,0,0,0,0,11,11,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,10,0,0,0,0,0,4,0] >;`

C62.30D6 in GAP, Magma, Sage, TeX

`C_6^2._{30}D_6`
`% in TeX`

`G:=Group("C6^2.30D6");`
`// GroupNames label`

`G:=SmallGroup(432,188);`
`// by ID`

`G=gap.SmallGroup(432,188);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,28,141,64,1124,4037,537]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^6=b^6=1,c^6=b^3,d^2=a^3,a*b=b*a,c*a*c^-1=a*b^2,d*a*d^-1=a^-1*b^4,b*c=c*b,b*d=d*b,d*c*d^-1=b^3*c^5>;`
`// generators/relations`

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