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## G = C62.16C32order 324 = 22·34

### 7th non-split extension by C62 of C32 acting via C32/C3=C3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C62.16C32
 Chief series C1 — C22 — C2×C6 — C62 — C32×A4 — C62.16C32
 Lower central C22 — C2×C6 — C62.16C32
 Upper central C1 — C32 — C3×C9

Generators and relations for C62.16C32
G = < a,b,c,d | a6=b6=c3=1, d3=a2, ab=ba, cac-1=ab3, ad=da, cbc-1=a3b4, bd=db, dcd-1=b2c >

Subgroups: 205 in 56 conjugacy classes, 21 normal (12 characteristic)
C1, C2, C3, C3, C3, C22, C6, C9, C32, C32, A4, C2×C6, C2×C6, C18, C3×C6, C3×C9, C3×C9, C33, C3.A4, C2×C18, C3×A4, C3×A4, C62, C3×C18, C32⋊C9, C3×C3.A4, C6×C18, C32×A4, C62.16C32
Quotients: C1, C3, C9, C32, A4, C3×C9, He3, 3- 1+2, C3×A4, C32⋊C9, C9×A4, C9⋊A4, C32⋊A4, C62.16C32

Smallest permutation representation of C62.16C32
On 108 points
Generators in S108
```(1 54 4 48 7 51)(2 46 5 49 8 52)(3 47 6 50 9 53)(10 29 13 32 16 35)(11 30 14 33 17 36)(12 31 15 34 18 28)(19 56 22 59 25 62)(20 57 23 60 26 63)(21 58 24 61 27 55)(37 76 40 79 43 73)(38 77 41 80 44 74)(39 78 42 81 45 75)(64 91 67 94 70 97)(65 92 68 95 71 98)(66 93 69 96 72 99)(82 107 85 101 88 104)(83 108 86 102 89 105)(84 100 87 103 90 106)
(1 42 63 12 90 66)(2 43 55 13 82 67)(3 44 56 14 83 68)(4 45 57 15 84 69)(5 37 58 16 85 70)(6 38 59 17 86 71)(7 39 60 18 87 72)(8 40 61 10 88 64)(9 41 62 11 89 65)(19 30 105 92 53 80)(20 31 106 93 54 81)(21 32 107 94 46 73)(22 33 108 95 47 74)(23 34 100 96 48 75)(24 35 101 97 49 76)(25 36 102 98 50 77)(26 28 103 99 51 78)(27 29 104 91 52 79)
(1 4 7)(2 58 88)(3 86 62)(5 61 82)(6 89 56)(8 55 85)(9 83 59)(10 97 104)(11 77 19)(12 28 54)(13 91 107)(14 80 22)(15 31 48)(16 94 101)(17 74 25)(18 34 51)(20 66 99)(21 43 29)(23 69 93)(24 37 32)(26 72 96)(27 40 35)(30 108 68)(33 102 71)(36 105 65)(38 95 50)(39 75 103)(41 98 53)(42 78 106)(44 92 47)(45 81 100)(46 67 79)(49 70 73)(52 64 76)(57 60 63)(84 87 90)
(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)```

`G:=sub<Sym(108)| (1,54,4,48,7,51)(2,46,5,49,8,52)(3,47,6,50,9,53)(10,29,13,32,16,35)(11,30,14,33,17,36)(12,31,15,34,18,28)(19,56,22,59,25,62)(20,57,23,60,26,63)(21,58,24,61,27,55)(37,76,40,79,43,73)(38,77,41,80,44,74)(39,78,42,81,45,75)(64,91,67,94,70,97)(65,92,68,95,71,98)(66,93,69,96,72,99)(82,107,85,101,88,104)(83,108,86,102,89,105)(84,100,87,103,90,106), (1,42,63,12,90,66)(2,43,55,13,82,67)(3,44,56,14,83,68)(4,45,57,15,84,69)(5,37,58,16,85,70)(6,38,59,17,86,71)(7,39,60,18,87,72)(8,40,61,10,88,64)(9,41,62,11,89,65)(19,30,105,92,53,80)(20,31,106,93,54,81)(21,32,107,94,46,73)(22,33,108,95,47,74)(23,34,100,96,48,75)(24,35,101,97,49,76)(25,36,102,98,50,77)(26,28,103,99,51,78)(27,29,104,91,52,79), (1,4,7)(2,58,88)(3,86,62)(5,61,82)(6,89,56)(8,55,85)(9,83,59)(10,97,104)(11,77,19)(12,28,54)(13,91,107)(14,80,22)(15,31,48)(16,94,101)(17,74,25)(18,34,51)(20,66,99)(21,43,29)(23,69,93)(24,37,32)(26,72,96)(27,40,35)(30,108,68)(33,102,71)(36,105,65)(38,95,50)(39,75,103)(41,98,53)(42,78,106)(44,92,47)(45,81,100)(46,67,79)(49,70,73)(52,64,76)(57,60,63)(84,87,90), (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)>;`

`G:=Group( (1,54,4,48,7,51)(2,46,5,49,8,52)(3,47,6,50,9,53)(10,29,13,32,16,35)(11,30,14,33,17,36)(12,31,15,34,18,28)(19,56,22,59,25,62)(20,57,23,60,26,63)(21,58,24,61,27,55)(37,76,40,79,43,73)(38,77,41,80,44,74)(39,78,42,81,45,75)(64,91,67,94,70,97)(65,92,68,95,71,98)(66,93,69,96,72,99)(82,107,85,101,88,104)(83,108,86,102,89,105)(84,100,87,103,90,106), (1,42,63,12,90,66)(2,43,55,13,82,67)(3,44,56,14,83,68)(4,45,57,15,84,69)(5,37,58,16,85,70)(6,38,59,17,86,71)(7,39,60,18,87,72)(8,40,61,10,88,64)(9,41,62,11,89,65)(19,30,105,92,53,80)(20,31,106,93,54,81)(21,32,107,94,46,73)(22,33,108,95,47,74)(23,34,100,96,48,75)(24,35,101,97,49,76)(25,36,102,98,50,77)(26,28,103,99,51,78)(27,29,104,91,52,79), (1,4,7)(2,58,88)(3,86,62)(5,61,82)(6,89,56)(8,55,85)(9,83,59)(10,97,104)(11,77,19)(12,28,54)(13,91,107)(14,80,22)(15,31,48)(16,94,101)(17,74,25)(18,34,51)(20,66,99)(21,43,29)(23,69,93)(24,37,32)(26,72,96)(27,40,35)(30,108,68)(33,102,71)(36,105,65)(38,95,50)(39,75,103)(41,98,53)(42,78,106)(44,92,47)(45,81,100)(46,67,79)(49,70,73)(52,64,76)(57,60,63)(84,87,90), (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) );`

`G=PermutationGroup([[(1,54,4,48,7,51),(2,46,5,49,8,52),(3,47,6,50,9,53),(10,29,13,32,16,35),(11,30,14,33,17,36),(12,31,15,34,18,28),(19,56,22,59,25,62),(20,57,23,60,26,63),(21,58,24,61,27,55),(37,76,40,79,43,73),(38,77,41,80,44,74),(39,78,42,81,45,75),(64,91,67,94,70,97),(65,92,68,95,71,98),(66,93,69,96,72,99),(82,107,85,101,88,104),(83,108,86,102,89,105),(84,100,87,103,90,106)], [(1,42,63,12,90,66),(2,43,55,13,82,67),(3,44,56,14,83,68),(4,45,57,15,84,69),(5,37,58,16,85,70),(6,38,59,17,86,71),(7,39,60,18,87,72),(8,40,61,10,88,64),(9,41,62,11,89,65),(19,30,105,92,53,80),(20,31,106,93,54,81),(21,32,107,94,46,73),(22,33,108,95,47,74),(23,34,100,96,48,75),(24,35,101,97,49,76),(25,36,102,98,50,77),(26,28,103,99,51,78),(27,29,104,91,52,79)], [(1,4,7),(2,58,88),(3,86,62),(5,61,82),(6,89,56),(8,55,85),(9,83,59),(10,97,104),(11,77,19),(12,28,54),(13,91,107),(14,80,22),(15,31,48),(16,94,101),(17,74,25),(18,34,51),(20,66,99),(21,43,29),(23,69,93),(24,37,32),(26,72,96),(27,40,35),(30,108,68),(33,102,71),(36,105,65),(38,95,50),(39,75,103),(41,98,53),(42,78,106),(44,92,47),(45,81,100),(46,67,79),(49,70,73),(52,64,76),(57,60,63),(84,87,90)], [(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)]])`

60 conjugacy classes

 class 1 2 3A ··· 3H 3I ··· 3N 6A ··· 6H 9A ··· 9F 9G ··· 9R 18A ··· 18R order 1 2 3 ··· 3 3 ··· 3 6 ··· 6 9 ··· 9 9 ··· 9 18 ··· 18 size 1 3 1 ··· 1 12 ··· 12 3 ··· 3 3 ··· 3 12 ··· 12 3 ··· 3

60 irreducible representations

 dim 1 1 1 1 1 3 3 3 3 3 3 3 type + + image C1 C3 C3 C3 C9 A4 He3 3- 1+2 C3×A4 C9×A4 C9⋊A4 C32⋊A4 kernel C62.16C32 C3×C3.A4 C6×C18 C32×A4 C3×A4 C3×C9 C2×C6 C2×C6 C32 C3 C3 C3 # reps 1 4 2 2 18 1 2 4 2 6 12 6

Matrix representation of C62.16C32 in GL4(𝔽19) generated by

 11 0 0 0 0 0 0 7 0 12 12 12 0 7 0 0
,
 1 0 0 0 0 0 11 0 0 11 0 0 0 8 8 8
,
 1 0 0 0 0 11 0 0 0 0 0 11 0 8 8 8
,
 9 0 0 0 0 7 15 6 0 9 1 13 0 10 4 11
`G:=sub<GL(4,GF(19))| [11,0,0,0,0,0,12,7,0,0,12,0,0,7,12,0],[1,0,0,0,0,0,11,8,0,11,0,8,0,0,0,8],[1,0,0,0,0,11,0,8,0,0,0,8,0,0,11,8],[9,0,0,0,0,7,9,10,0,15,1,4,0,6,13,11] >;`

C62.16C32 in GAP, Magma, Sage, TeX

`C_6^2._{16}C_3^2`
`% in TeX`

`G:=Group("C6^2.16C3^2");`
`// GroupNames label`

`G:=SmallGroup(324,52);`
`// by ID`

`G=gap.SmallGroup(324,52);`
`# by ID`

`G:=PCGroup([6,-3,-3,-3,-3,-2,2,361,43,4864,8753]);`
`// Polycyclic`

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

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