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## G = C12.84S32order 432 = 24·33

### 4th 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 — C12.84S32
 Chief series C1 — C3 — C32 — He3 — C2×He3 — C2×C32⋊C6 — C6.S32 — C12.84S32
 Lower central He3 — C2×He3 — C12.84S32
 Upper central C1 — C2 — C4

Generators and relations for C12.84S32
G = < a,b,c,d,e | a12=b3=d3=e2=1, c2=a6, ab=ba, cac-1=eae=a-1, ad=da, cbc-1=b-1, dbd-1=a8b, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 927 in 156 conjugacy classes, 35 normal (23 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×C12, C3×C12, S3×C6, C2×C3⋊S3, C4○D12, D42S3, Q83S3, C32⋊C6, He3⋊C2, C2×He3, S3×Dic3, C6.D6, D6⋊S3, C3⋊D12, C3×Dic6, S3×C12, C3×D12, C324Q8, C12⋊S3, C32⋊C12, He33C4, C4×He3, C2×C32⋊C6, C2×He3⋊C2, D125S3, D6.6D6, C6.S32, He33D4, He33Q8, He34D4, C4×He3⋊C2, C12.84S32
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C22×S3, S32, D42S3, Q83S3, C2×S32, C32⋊D6, D12⋊S3, C2×C32⋊D6, C12.84S32

Smallest permutation representation of C12.84S32
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)
(1 21 26)(2 22 27)(3 23 28)(4 24 29)(5 13 30)(6 14 31)(7 15 32)(8 16 33)(9 17 34)(10 18 35)(11 19 36)(12 20 25)(37 56 62)(38 57 63)(39 58 64)(40 59 65)(41 60 66)(42 49 67)(43 50 68)(44 51 69)(45 52 70)(46 53 71)(47 54 72)(48 55 61)
(1 50 7 56)(2 49 8 55)(3 60 9 54)(4 59 10 53)(5 58 11 52)(6 57 12 51)(13 39 19 45)(14 38 20 44)(15 37 21 43)(16 48 22 42)(17 47 23 41)(18 46 24 40)(25 69 31 63)(26 68 32 62)(27 67 33 61)(28 66 34 72)(29 65 35 71)(30 64 36 70)
(13 17 21)(14 18 22)(15 19 23)(16 20 24)(25 33 29)(26 34 30)(27 35 31)(28 36 32)(37 45 41)(38 46 42)(39 47 43)(40 48 44)(61 65 69)(62 66 70)(63 67 71)(64 68 72)
(1 53)(2 52)(3 51)(4 50)(5 49)(6 60)(7 59)(8 58)(9 57)(10 56)(11 55)(12 54)(13 67)(14 66)(15 65)(16 64)(17 63)(18 62)(19 61)(20 72)(21 71)(22 70)(23 69)(24 68)(25 47)(26 46)(27 45)(28 44)(29 43)(30 42)(31 41)(32 40)(33 39)(34 38)(35 37)(36 48)```

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

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

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

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 2 2 12 12 12 12 18 18 36 36

32 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 4 4 4 4 4 6 6 6 type + + + + + + + + + + + + - + + + + image C1 C2 C2 C2 C2 C2 S3 S3 D6 D6 D6 C4○D4 S32 D4⋊2S3 Q8⋊3S3 C2×S32 D12⋊S3 C32⋊D6 C2×C32⋊D6 C12.84S32 kernel C12.84S32 C6.S32 He3⋊3D4 He3⋊3Q8 He3⋊4D4 C4×He3⋊C2 C32⋊4Q8 C12⋊S3 C3⋊Dic3 C3×C12 C2×C3⋊S3 He3 C12 C32 C32 C6 C3 C4 C2 C1 # reps 1 2 2 1 1 1 1 1 2 2 2 2 1 1 1 1 2 2 2 4

Matrix representation of C12.84S32 in GL6(𝔽13)

 6 0 0 0 0 0 0 6 0 0 0 0 0 0 6 0 0 0 0 0 0 11 0 0 0 0 0 0 11 0 0 0 0 0 0 11
,
 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1 0
,
 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0
,
 1 0 0 0 0 0 0 3 0 0 0 0 0 0 9 0 0 0 0 0 0 1 0 0 0 0 0 0 3 0 0 0 0 0 0 9
,
 0 0 0 5 0 0 0 0 0 0 0 5 0 0 0 0 5 0 8 0 0 0 0 0 0 0 8 0 0 0 0 8 0 0 0 0

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

C12.84S32 in GAP, Magma, Sage, TeX

`C_{12}._{84}S_3^2`
`% in TeX`

`G:=Group("C12.84S3^2");`
`// GroupNames label`

`G:=SmallGroup(432,296);`
`// by ID`

`G=gap.SmallGroup(432,296);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,64,254,135,58,571,4037,537,14118,7069]);`
`// 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=e*a*e=a^-1,a*d=d*a,c*b*c^-1=b^-1,d*b*d^-1=a^8*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;`
`// generators/relations`

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