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## G = C62.19D6order 432 = 24·33

### 2nd non-split extension by C62 of D6 acting via D6/C2=S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C62.19D6
 Chief series C1 — C3 — C32 — C3×C6 — C62 — C22×He3 — C2×C32⋊C12 — C62.19D6
 Lower central C32 — C3×C6 — C62.19D6
 Upper central C1 — C22 — C2×C4

Generators and relations for C62.19D6
G = < a,b,c,d | a6=b6=1, c6=d2=a3, ab=ba, cac-1=dad-1=ab2, bc=cb, dbd-1=b-1, dcd-1=b3c5 >

Subgroups: 397 in 107 conjugacy classes, 40 normal (36 characteristic)
C1, C2, C3, C3, C4, C22, C6, C6, C2×C4, C2×C4, C32, C32, Dic3, C12, C2×C6, C2×C6, C4⋊C4, C3×C6, C3×C6, C2×Dic3, C2×C12, C2×C12, He3, C3×Dic3, C3⋊Dic3, C3⋊Dic3, C3×C12, C62, C62, Dic3⋊C4, C3×C4⋊C4, C2×He3, C6×Dic3, C2×C3⋊Dic3, C6×C12, C6×C12, C32⋊C12, C32⋊C12, C4×He3, C22×He3, C3×Dic3⋊C4, C6.Dic6, C2×C32⋊C12, C2×C4×He3, C62.19D6
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, D4, Q8, C12, D6, C2×C6, C4⋊C4, C3×S3, Dic6, C4×S3, C3⋊D4, C2×C12, C3×D4, C3×Q8, S3×C6, Dic3⋊C4, C3×C4⋊C4, C32⋊C6, C3×Dic6, S3×C12, C3×C3⋊D4, C2×C32⋊C6, C3×Dic3⋊C4, He33Q8, C4×C32⋊C6, He36D4, C62.19D6

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

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

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

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

62 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 3F 4A 4B 4C 4D 4E 4F 6A 6B 6C 6D ··· 6I 6J ··· 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 ··· 6 6 ··· 6 12 12 12 12 12 ··· 12 12 ··· 12 size 1 1 1 1 2 3 3 6 6 6 2 2 18 18 18 18 2 2 2 3 ··· 3 6 ··· 6 2 2 2 2 6 ··· 6 18 ··· 18

62 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 6 6 6 6 6 type + + + + + - + - + + - image C1 C2 C2 C3 C4 C6 C6 C12 S3 D4 Q8 D6 C3×S3 Dic6 C4×S3 C3⋊D4 C3×D4 C3×Q8 S3×C6 C3×Dic6 S3×C12 C3×C3⋊D4 C32⋊C6 C2×C32⋊C6 He3⋊3Q8 C4×C32⋊C6 He3⋊6D4 kernel C62.19D6 C2×C32⋊C12 C2×C4×He3 C6.Dic6 C32⋊C12 C2×C3⋊Dic3 C6×C12 C3⋊Dic3 C6×C12 C2×He3 C2×He3 C62 C2×C12 C3×C6 C3×C6 C3×C6 C3×C6 C3×C6 C2×C6 C6 C6 C6 C2×C4 C22 C2 C2 C2 # reps 1 2 1 2 4 4 2 8 1 1 1 1 2 2 2 2 2 2 2 4 4 4 1 1 2 2 2

Matrix representation of C62.19D6 in GL10(𝔽13)

 4 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 2 2 2 12 12 2 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1
,
 12 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 8 0 0 3 0 0 0 0 0 0 0 8 0 0 3 0 0 0 0 0 11 11 11 0 0 3
,
 7 10 0 0 0 0 0 0 0 0 3 10 0 0 0 0 0 0 0 0 0 0 4 2 0 0 0 0 0 0 0 0 11 2 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 8 0 0 3 0 0 0 0 0 8 4 2 9 10 9
,
 2 11 0 0 0 0 0 0 0 0 9 11 0 0 0 0 0 0 0 0 0 0 10 7 0 0 0 0 0 0 0 0 10 3 0 0 0 0 0 0 0 0 0 0 6 0 0 2 0 0 0 0 0 0 0 5 0 0 6 0 0 0 0 0 10 10 12 8 8 10 0 0 0 0 2 0 0 7 0 0 0 0 0 0 0 9 0 0 8 0 0 0 0 0 9 7 0 1 4 1

`G:=sub<GL(10,GF(13))| [4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,2,0,0,0,0,0,0,0,0,1,2,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,2,0,1],[12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,9,0,0,8,0,11,0,0,0,0,0,9,0,0,8,11,0,0,0,0,0,0,9,0,0,11,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,3],[7,3,0,0,0,0,0,0,0,0,10,10,0,0,0,0,0,0,0,0,0,0,4,11,0,0,0,0,0,0,0,0,2,2,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,8,0,0,0,0,0,9,0,0,8,4,0,0,0,0,0,0,3,0,0,2,0,0,0,0,0,0,0,1,0,9,0,0,0,0,0,0,0,0,3,10,0,0,0,0,0,0,0,0,0,9],[2,9,0,0,0,0,0,0,0,0,11,11,0,0,0,0,0,0,0,0,0,0,10,10,0,0,0,0,0,0,0,0,7,3,0,0,0,0,0,0,0,0,0,0,6,0,10,2,0,9,0,0,0,0,0,5,10,0,9,7,0,0,0,0,0,0,12,0,0,0,0,0,0,0,2,0,8,7,0,1,0,0,0,0,0,6,8,0,8,4,0,0,0,0,0,0,10,0,0,1] >;`

C62.19D6 in GAP, Magma, Sage, TeX

`C_6^2._{19}D_6`
`% in TeX`

`G:=Group("C6^2.19D6");`
`// GroupNames label`

`G:=SmallGroup(432,139);`
`// by ID`

`G=gap.SmallGroup(432,139);`
`# by ID`

`G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,168,365,92,4037,2035,14118]);`
`// Polycyclic`

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

׿
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