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## G = C12.S32order 432 = 24·33

### 23rd non-split extension by C12 of S32 acting via S32/C32=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C2×He3 — C12.S32
 Chief series C1 — C3 — C32 — He3 — C2×He3 — C2×C32⋊C6 — He3⋊3D4 — C12.S32
 Lower central He3 — C2×He3 — C12.S32
 Upper central C1 — C2 — C4

Generators and relations for C12.S32
G = < a,b,c,d,e | a12=b3=d3=e2=1, c2=a6, ab=ba, cac-1=a-1, ad=da, eae=a5, cbc-1=b-1, dbd-1=a4b, be=eb, cd=dc, ece=a6c, ede=d-1 >

Subgroups: 943 in 156 conjugacy classes, 35 normal (25 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C2×C4, D4, Q8, C32, C32, Dic3, C12, C12, D6, C2×C6, C4○D4, C3×S3, C3⋊S3, C3×C6, C3×C6, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C3×Q8, He3, C3×Dic3, C3⋊Dic3, C3⋊Dic3, C3×C12, C3×C12, S3×C6, C2×C3⋊S3, C4○D12, D42S3, Q83S3, C32⋊C6, He3⋊C2, C2×He3, S3×Dic3, D6⋊S3, C3⋊D12, C3×Dic6, S3×C12, C3×D12, C324Q8, C4×C3⋊S3, C32⋊C12, C32⋊C12, C4×He3, C2×C32⋊C6, C2×He3⋊C2, D125S3, D12⋊S3, He3⋊(C2×C4), He33D4, He33Q8, C4×C32⋊C6, He35D4, C12.S32
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C22×S3, S32, C4○D12, Q83S3, C2×S32, C32⋊D6, D6.6D6, C2×C32⋊D6, C12.S32

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

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

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

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

32 conjugacy classes

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

32 irreducible representations

 dim 1 1 1 1 1 1 12 2 2 2 2 2 2 2 4 4 4 4 6 6 type + + + + + + - + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C12.S32 S3 S3 D6 D6 D6 C4○D4 C4○D12 S32 Q8⋊3S3 C2×S32 D6.6D6 C32⋊D6 C2×C32⋊D6 kernel C12.S32 He3⋊(C2×C4) He3⋊3D4 He3⋊3Q8 C4×C32⋊C6 He3⋊5D4 C1 C32⋊4Q8 C4×C3⋊S3 C3⋊Dic3 C3×C12 C2×C3⋊S3 He3 C32 C12 C32 C6 C3 C4 C2 # reps 1 2 2 1 1 1 1 1 1 3 2 1 2 4 1 1 1 2 2 2

Matrix representation of C12.S32 in GL10(𝔽13)

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

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

C12.S32 in GAP, Magma, Sage, TeX

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

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

`G:=SmallGroup(432,299);`
`// by ID`

`G=gap.SmallGroup(432,299);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,141,64,135,58,571,4037,537,14118,7069]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^12=b^3=d^3=e^2=1,c^2=a^6,a*b=b*a,c*a*c^-1=a^-1,a*d=d*a,e*a*e=a^5,c*b*c^-1=b^-1,d*b*d^-1=a^4*b,b*e=e*b,c*d=d*c,e*c*e=a^6*c,e*d*e=d^-1>;`
`// generators/relations`

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