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## G = C12.40S32order 432 = 24·33

### 40th non-split extension by C12 of S32 acting via S32/C32=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32×C6 — C12.40S32
 Chief series C1 — C3 — C32 — C33 — C32×C6 — C32×Dic3 — C33⋊8(C2×C4) — C12.40S32
 Lower central C33 — C32×C6 — C12.40S32
 Upper central C1 — C2 — C4

Generators and relations for C12.40S32
G = < a,b,c,d,e,f | a3=b3=c3=d4=e2=f2=1, ab=ba, ac=ca, ad=da, eae=faf=a-1, bc=cb, bd=db, ebe=fbf=b-1, dcd-1=ece=fcf=c-1, ede=d-1, df=fd, fef=d2e >

Subgroups: 2104 in 304 conjugacy classes, 68 normal (22 characteristic)
C1, C2, C2, C3, C3, C3, C4, C4, C22, S3, C6, C6, C6, C2×C4, D4, Q8, C32, C32, C32, Dic3, Dic3, C12, C12, C12, D6, C2×C6, C4○D4, C3×S3, C3⋊S3, C3×C6, C3×C6, C3×C6, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C3×Q8, C33, C3×Dic3, C3×Dic3, C3⋊Dic3, C3×C12, C3×C12, C3×C12, S3×C6, C2×C3⋊S3, C2×C3⋊S3, C4○D12, Q83S3, C3×C3⋊S3, C33⋊C2, C32×C6, C6.D6, C3⋊D12, C3×Dic6, S3×C12, C4×C3⋊S3, C4×C3⋊S3, C12⋊S3, Q8×C32, C32×Dic3, C3×C3⋊Dic3, C32×C12, C6×C3⋊S3, C2×C33⋊C2, D6.6D6, C12.26D6, C338(C2×C4), C338D4, C32×Dic6, C12×C3⋊S3, C3312D4, C12.40S32
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C3⋊S3, C22×S3, S32, C2×C3⋊S3, C4○D12, Q83S3, C2×S32, C22×C3⋊S3, S3×C3⋊S3, D6.6D6, C12.26D6, C2×S3×C3⋊S3, C12.40S32

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

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

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

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

54 conjugacy classes

 class 1 2A 2B 2C 2D 3A ··· 3E 3F 3G 3H 3I 4A 4B 4C 4D 4E 6A ··· 6E 6F 6G 6H 6I 6J 6K 12A 12B 12C ··· 12N 12O ··· 12V 12W 12X order 1 2 2 2 2 3 ··· 3 3 3 3 3 4 4 4 4 4 6 ··· 6 6 6 6 6 6 6 12 12 12 ··· 12 12 ··· 12 12 12 size 1 1 18 54 54 2 ··· 2 4 4 4 4 2 6 6 9 9 2 ··· 2 4 4 4 4 18 18 2 2 4 ··· 4 12 ··· 12 18 18

54 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 S3 S3 D6 D6 D6 D6 C4○D4 C4○D12 S32 Q8⋊3S3 C2×S32 D6.6D6 kernel C12.40S32 C33⋊8(C2×C4) C33⋊8D4 C32×Dic6 C12×C3⋊S3 C33⋊12D4 C3×Dic6 C4×C3⋊S3 C3×Dic3 C3⋊Dic3 C3×C12 C2×C3⋊S3 C33 C32 C12 C32 C6 C3 # reps 1 2 2 1 1 1 4 1 8 1 5 1 2 4 4 4 4 8

Matrix representation of C12.40S32 in GL8(𝔽13)

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

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

C12.40S32 in GAP, Magma, Sage, TeX

`C_{12}._{40}S_3^2`
`% in TeX`

`G:=Group("C12.40S3^2");`
`// GroupNames label`

`G:=SmallGroup(432,665);`
`// by ID`

`G=gap.SmallGroup(432,665);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,56,64,254,58,571,2028,14118]);`
`// Polycyclic`

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

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