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## G = C12.69S32order 432 = 24·33

### 26th non-split extension by C12 of S32 acting via S32/C3×S3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C33 — C12.69S32
 Chief series C1 — C3 — C32 — C33 — C32×C6 — C32×C12 — C32×C3⋊C8 — C12.69S32
 Lower central C33 — C12.69S32
 Upper central C1 — C4

Generators and relations for C12.69S32
G = < a,b,c,d,e | a3=b3=c8=d3=e2=1, ab=ba, ac=ca, ad=da, eae=a-1, cbc-1=ebe=b-1, bd=db, cd=dc, ce=ec, ede=d-1 >

Subgroups: 1160 in 196 conjugacy classes, 50 normal (20 characteristic)
C1, C2, C2 [×2], C3, C3 [×4], C3 [×4], C4, C4, C22, S3 [×18], C6, C6 [×4], C6 [×4], C8 [×2], C2×C4, C32, C32 [×4], C32 [×4], Dic3 [×9], C12, C12 [×4], C12 [×4], D6 [×9], C2×C8, C3⋊S3 [×18], C3×C6, C3×C6 [×4], C3×C6 [×4], C3⋊C8, C3⋊C8 [×4], C24 [×5], C4×S3 [×9], C33, C3⋊Dic3 [×9], C3×C12, C3×C12 [×4], C3×C12 [×4], C2×C3⋊S3 [×9], S3×C8 [×5], C33⋊C2 [×2], C32×C6, C3×C3⋊C8 [×4], C3×C3⋊C8 [×4], C324C8, C3×C24, C4×C3⋊S3 [×9], C335C4, C32×C12, C2×C33⋊C2, C12.29D6 [×4], C8×C3⋊S3, C32×C3⋊C8, C3×C324C8, C4×C33⋊C2, C12.69S32
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×5], C8 [×2], C2×C4, D6 [×5], C2×C8, C3⋊S3, C4×S3 [×5], S32 [×4], C2×C3⋊S3, S3×C8 [×5], C6.D6 [×4], C4×C3⋊S3, S3×C3⋊S3, C12.29D6 [×4], C8×C3⋊S3, C338(C2×C4), C12.69S32

Smallest permutation representation of C12.69S32
On 72 points
Generators in S72
```(1 31 68)(2 32 69)(3 25 70)(4 26 71)(5 27 72)(6 28 65)(7 29 66)(8 30 67)(9 22 56)(10 23 49)(11 24 50)(12 17 51)(13 18 52)(14 19 53)(15 20 54)(16 21 55)(33 62 45)(34 63 46)(35 64 47)(36 57 48)(37 58 41)(38 59 42)(39 60 43)(40 61 44)
(1 36 11)(2 12 37)(3 38 13)(4 14 39)(5 40 15)(6 16 33)(7 34 9)(8 10 35)(17 58 32)(18 25 59)(19 60 26)(20 27 61)(21 62 28)(22 29 63)(23 64 30)(24 31 57)(41 69 51)(42 52 70)(43 71 53)(44 54 72)(45 65 55)(46 56 66)(47 67 49)(48 50 68)
(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)
(1 24 48)(2 17 41)(3 18 42)(4 19 43)(5 20 44)(6 21 45)(7 22 46)(8 23 47)(9 63 66)(10 64 67)(11 57 68)(12 58 69)(13 59 70)(14 60 71)(15 61 72)(16 62 65)(25 52 38)(26 53 39)(27 54 40)(28 55 33)(29 56 34)(30 49 35)(31 50 36)(32 51 37)
(9 34)(10 35)(11 36)(12 37)(13 38)(14 39)(15 40)(16 33)(17 41)(18 42)(19 43)(20 44)(21 45)(22 46)(23 47)(24 48)(25 70)(26 71)(27 72)(28 65)(29 66)(30 67)(31 68)(32 69)(49 64)(50 57)(51 58)(52 59)(53 60)(54 61)(55 62)(56 63)```

`G:=sub<Sym(72)| (1,31,68)(2,32,69)(3,25,70)(4,26,71)(5,27,72)(6,28,65)(7,29,66)(8,30,67)(9,22,56)(10,23,49)(11,24,50)(12,17,51)(13,18,52)(14,19,53)(15,20,54)(16,21,55)(33,62,45)(34,63,46)(35,64,47)(36,57,48)(37,58,41)(38,59,42)(39,60,43)(40,61,44), (1,36,11)(2,12,37)(3,38,13)(4,14,39)(5,40,15)(6,16,33)(7,34,9)(8,10,35)(17,58,32)(18,25,59)(19,60,26)(20,27,61)(21,62,28)(22,29,63)(23,64,30)(24,31,57)(41,69,51)(42,52,70)(43,71,53)(44,54,72)(45,65,55)(46,56,66)(47,67,49)(48,50,68), (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), (1,24,48)(2,17,41)(3,18,42)(4,19,43)(5,20,44)(6,21,45)(7,22,46)(8,23,47)(9,63,66)(10,64,67)(11,57,68)(12,58,69)(13,59,70)(14,60,71)(15,61,72)(16,62,65)(25,52,38)(26,53,39)(27,54,40)(28,55,33)(29,56,34)(30,49,35)(31,50,36)(32,51,37), (9,34)(10,35)(11,36)(12,37)(13,38)(14,39)(15,40)(16,33)(17,41)(18,42)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48)(25,70)(26,71)(27,72)(28,65)(29,66)(30,67)(31,68)(32,69)(49,64)(50,57)(51,58)(52,59)(53,60)(54,61)(55,62)(56,63)>;`

`G:=Group( (1,31,68)(2,32,69)(3,25,70)(4,26,71)(5,27,72)(6,28,65)(7,29,66)(8,30,67)(9,22,56)(10,23,49)(11,24,50)(12,17,51)(13,18,52)(14,19,53)(15,20,54)(16,21,55)(33,62,45)(34,63,46)(35,64,47)(36,57,48)(37,58,41)(38,59,42)(39,60,43)(40,61,44), (1,36,11)(2,12,37)(3,38,13)(4,14,39)(5,40,15)(6,16,33)(7,34,9)(8,10,35)(17,58,32)(18,25,59)(19,60,26)(20,27,61)(21,62,28)(22,29,63)(23,64,30)(24,31,57)(41,69,51)(42,52,70)(43,71,53)(44,54,72)(45,65,55)(46,56,66)(47,67,49)(48,50,68), (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), (1,24,48)(2,17,41)(3,18,42)(4,19,43)(5,20,44)(6,21,45)(7,22,46)(8,23,47)(9,63,66)(10,64,67)(11,57,68)(12,58,69)(13,59,70)(14,60,71)(15,61,72)(16,62,65)(25,52,38)(26,53,39)(27,54,40)(28,55,33)(29,56,34)(30,49,35)(31,50,36)(32,51,37), (9,34)(10,35)(11,36)(12,37)(13,38)(14,39)(15,40)(16,33)(17,41)(18,42)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48)(25,70)(26,71)(27,72)(28,65)(29,66)(30,67)(31,68)(32,69)(49,64)(50,57)(51,58)(52,59)(53,60)(54,61)(55,62)(56,63) );`

`G=PermutationGroup([(1,31,68),(2,32,69),(3,25,70),(4,26,71),(5,27,72),(6,28,65),(7,29,66),(8,30,67),(9,22,56),(10,23,49),(11,24,50),(12,17,51),(13,18,52),(14,19,53),(15,20,54),(16,21,55),(33,62,45),(34,63,46),(35,64,47),(36,57,48),(37,58,41),(38,59,42),(39,60,43),(40,61,44)], [(1,36,11),(2,12,37),(3,38,13),(4,14,39),(5,40,15),(6,16,33),(7,34,9),(8,10,35),(17,58,32),(18,25,59),(19,60,26),(20,27,61),(21,62,28),(22,29,63),(23,64,30),(24,31,57),(41,69,51),(42,52,70),(43,71,53),(44,54,72),(45,65,55),(46,56,66),(47,67,49),(48,50,68)], [(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)], [(1,24,48),(2,17,41),(3,18,42),(4,19,43),(5,20,44),(6,21,45),(7,22,46),(8,23,47),(9,63,66),(10,64,67),(11,57,68),(12,58,69),(13,59,70),(14,60,71),(15,61,72),(16,62,65),(25,52,38),(26,53,39),(27,54,40),(28,55,33),(29,56,34),(30,49,35),(31,50,36),(32,51,37)], [(9,34),(10,35),(11,36),(12,37),(13,38),(14,39),(15,40),(16,33),(17,41),(18,42),(19,43),(20,44),(21,45),(22,46),(23,47),(24,48),(25,70),(26,71),(27,72),(28,65),(29,66),(30,67),(31,68),(32,69),(49,64),(50,57),(51,58),(52,59),(53,60),(54,61),(55,62),(56,63)])`

72 conjugacy classes

 class 1 2A 2B 2C 3A ··· 3E 3F 3G 3H 3I 4A 4B 4C 4D 6A ··· 6E 6F 6G 6H 6I 8A 8B 8C 8D 8E 8F 8G 8H 12A ··· 12J 12K ··· 12R 24A ··· 24P 24Q 24R 24S 24T order 1 2 2 2 3 ··· 3 3 3 3 3 4 4 4 4 6 ··· 6 6 6 6 6 8 8 8 8 8 8 8 8 12 ··· 12 12 ··· 12 24 ··· 24 24 24 24 24 size 1 1 27 27 2 ··· 2 4 4 4 4 1 1 27 27 2 ··· 2 4 4 4 4 3 3 3 3 9 9 9 9 2 ··· 2 4 ··· 4 6 ··· 6 18 18 18 18

72 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 4 4 4 type + + + + + + + + + image C1 C2 C2 C2 C4 C4 C8 S3 S3 D6 C4×S3 S3×C8 S32 C6.D6 C12.29D6 kernel C12.69S32 C32×C3⋊C8 C3×C32⋊4C8 C4×C33⋊C2 C33⋊5C4 C2×C33⋊C2 C33⋊C2 C3×C3⋊C8 C32⋊4C8 C3×C12 C3×C6 C32 C12 C6 C3 # reps 1 1 1 1 2 2 8 4 1 5 10 20 4 4 8

Matrix representation of C12.69S32 in GL6(𝔽73)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 72 0 0 0 0 1 72 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 72 0 0 0 0 1 72
,
 22 0 0 0 0 0 0 22 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 72 72 0 0 0 0 1 0 0 0 0 0 0 0 0 72 0 0 0 0 1 72 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 72 72 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0

`G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,72,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,72,72],[22,0,0,0,0,0,0,22,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[72,1,0,0,0,0,72,0,0,0,0,0,0,0,0,1,0,0,0,0,72,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,72,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;`

C12.69S32 in GAP, Magma, Sage, TeX

`C_{12}._{69}S_3^2`
`% in TeX`

`G:=Group("C12.69S3^2");`
`// GroupNames label`

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

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

`G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,28,36,58,571,2028,14118]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^3=b^3=c^8=d^3=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e=a^-1,c*b*c^-1=e*b*e=b^-1,b*d=d*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;`
`// generators/relations`

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