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## G = C12.85S32order 432 = 24·33

### 5th non-split extension by C12 of S32 acting via S32/C3⋊S3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C2×He3 — C12.85S32
 Chief series C1 — C3 — C32 — He3 — C2×He3 — C32⋊C12 — He3⋊(C2×C4) — C12.85S32
 Lower central He3 — C2×He3 — C12.85S32
 Upper central C1 — C2 — C4

Generators and relations for C12.85S32
G = < a,b,c,d,e,f | a3=b3=c3=d2=e4=1, f2=e2, ab=ba, cac-1=ab-1, dad=eae-1=faf-1=a-1, bc=cb, bd=db, be=eb, fbf-1=b-1, dcd=ece-1=c-1, cf=fc, de=ed, df=fd, fef-1=e-1 >

Subgroups: 787 in 149 conjugacy classes, 37 normal (13 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C2×C4, Q8, C32, C32, Dic3, C12, C12, D6, C2×C6, C2×Q8, C3×S3, C3×C6, C3×C6, Dic6, C4×S3, C2×Dic3, C2×C12, C3×Q8, He3, C3×Dic3, C3⋊Dic3, C3×C12, C3×C12, S3×C6, C2×Dic6, S3×Q8, He3⋊C2, C2×He3, S3×Dic3, C322Q8, C3×Dic6, S3×C12, C324Q8, C32⋊C12, He33C4, C4×He3, C2×He3⋊C2, S3×Dic6, He32Q8, He3⋊(C2×C4), He33Q8, C4×He3⋊C2, C12.85S32
Quotients: C1, C2, C22, S3, Q8, C23, D6, C2×Q8, C22×S3, S32, S3×Q8, C2×S32, C32⋊D6, Dic3.D6, C2×C32⋊D6, C12.85S32

Smallest permutation representation of C12.85S32
On 72 points
Generators in S72
```(1 8 45)(2 46 5)(3 6 47)(4 48 7)(9 66 43)(10 44 67)(11 68 41)(12 42 65)(13 21 37)(14 38 22)(15 23 39)(16 40 24)(17 60 62)(18 63 57)(19 58 64)(20 61 59)(25 49 71)(26 72 50)(27 51 69)(28 70 52)(29 55 36)(30 33 56)(31 53 34)(32 35 54)
(1 30 16)(2 31 13)(3 32 14)(4 29 15)(5 34 37)(6 35 38)(7 36 39)(8 33 40)(9 69 63)(10 70 64)(11 71 61)(12 72 62)(17 42 50)(18 43 51)(19 44 52)(20 41 49)(21 46 53)(22 47 54)(23 48 55)(24 45 56)(25 59 68)(26 60 65)(27 57 66)(28 58 67)
(5 37 34)(6 35 38)(7 39 36)(8 33 40)(9 69 63)(10 64 70)(11 71 61)(12 62 72)(21 46 53)(22 54 47)(23 48 55)(24 56 45)(25 68 59)(26 60 65)(27 66 57)(28 58 67)
(1 3)(2 4)(5 48)(6 45)(7 46)(8 47)(9 68)(10 65)(11 66)(12 67)(13 15)(14 16)(17 19)(18 20)(21 39)(22 40)(23 37)(24 38)(25 69)(26 70)(27 71)(28 72)(29 31)(30 32)(33 54)(34 55)(35 56)(36 53)(41 43)(42 44)(49 51)(50 52)(57 61)(58 62)(59 63)(60 64)
(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 18 3 20)(2 17 4 19)(5 60 7 58)(6 59 8 57)(9 22 11 24)(10 21 12 23)(13 42 15 44)(14 41 16 43)(25 33 27 35)(26 36 28 34)(29 52 31 50)(30 51 32 49)(37 65 39 67)(38 68 40 66)(45 63 47 61)(46 62 48 64)(53 72 55 70)(54 71 56 69)```

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

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

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

32 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 4A 4B ··· 4F 6A 6B 6C 6D 6E 6F 12A 12B 12C 12D 12E 12F 12G 12H 12I 12J 12K 12L order 1 2 2 2 3 3 3 3 4 4 ··· 4 6 6 6 6 6 6 12 12 12 12 12 12 12 12 12 12 12 12 size 1 1 9 9 2 6 6 12 2 18 ··· 18 2 6 6 12 18 18 2 2 12 12 12 12 18 18 36 36 36 36

32 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 4 4 4 4 6 6 6 type + + + + + + - + + + - + + + - image C1 C2 C2 C2 C2 S3 Q8 D6 D6 S32 S3×Q8 C2×S32 Dic3.D6 C32⋊D6 C2×C32⋊D6 C12.85S32 kernel C12.85S32 He3⋊2Q8 He3⋊(C2×C4) He3⋊3Q8 C4×He3⋊C2 C32⋊4Q8 He3⋊C2 C3⋊Dic3 C3×C12 C12 C32 C6 C3 C4 C2 C1 # reps 1 2 2 2 1 2 2 4 2 1 2 1 2 2 2 4

Matrix representation of C12.85S32 in GL6(𝔽13)

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

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

C12.85S32 in GAP, Magma, Sage, TeX

`C_{12}._{85}S_3^2`
`% in TeX`

`G:=Group("C12.85S3^2");`
`// GroupNames label`

`G:=SmallGroup(432,298);`
`// by ID`

`G=gap.SmallGroup(432,298);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,56,64,254,135,58,571,4037,537,14118,7069]);`
`// Polycyclic`

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

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