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## G = C12.58S32order 432 = 24·33

### 15th non-split extension by C12 of S32 acting via S32/C3×S3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32×C6 — C12.58S32
 Chief series C1 — C3 — C32 — C33 — C32×C6 — S3×C3×C6 — C33⋊7D4 — C12.58S32
 Lower central C33 — C32×C6 — C12.58S32
 Upper central C1 — C2 — C4

Generators and relations for C12.58S32
G = < a,b,c,d,e,f | a3=b3=c3=d4=e2=f2=1, ab=ba, ac=ca, dad-1=eae=faf=a-1, bc=cb, dbd-1=ebe=fbf=b-1, cd=dc, ece=fcf=c-1, ede=d-1, df=fd, fef=d2e >

Subgroups: 2064 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, D6, C2×C6, C4○D4, C3×S3, C3⋊S3, C3×C6, C3×C6, C3×C6, Dic6, C4×S3, 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, C62, C4○D12, Q83S3, S3×C32, C33⋊C2, C32×C6, C6.D6, C3⋊D12, C3×Dic6, S3×C12, C324Q8, C4×C3⋊S3, C12⋊S3, C327D4, C6×C12, C32×Dic3, C3×C3⋊Dic3, C32×C12, S3×C3×C6, C2×C33⋊C2, D6.6D6, C12.59D6, C338(C2×C4), C337D4, S3×C3×C12, C3×C324Q8, C3312D4, C12.58S32
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, C12.59D6, C2×S3×C3⋊S3, C12.58S32

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

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

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

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

63 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 ··· 6Q 12A ··· 12H 12I ··· 12Q 12R ··· 12Y 12Z 12AA 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 6 54 54 2 ··· 2 4 4 4 4 2 3 3 18 18 2 ··· 2 4 4 4 4 6 ··· 6 2 ··· 2 4 ··· 4 6 ··· 6 36 36

63 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 C12.58S32 C33⋊8(C2×C4) C33⋊7D4 S3×C3×C12 C3×C32⋊4Q8 C33⋊12D4 S3×C12 C32⋊4Q8 C3×Dic3 C3⋊Dic3 C3×C12 S3×C6 C33 C32 C12 C32 C6 C3 # reps 1 2 2 1 1 1 4 1 4 2 5 4 2 16 4 1 4 8

Matrix representation of C12.58S32 in GL6(𝔽13)

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

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

C12.58S32 in GAP, Magma, Sage, TeX

`C_{12}._{58}S_3^2`
`% in TeX`

`G:=Group("C12.58S3^2");`
`// GroupNames label`

`G:=SmallGroup(432,669);`
`// by ID`

`G=gap.SmallGroup(432,669);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,141,64,135,58,571,2028,14118]);`
`// 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,d*a*d^-1=e*a*e=f*a*f=a^-1,b*c=c*b,d*b*d^-1=e*b*e=f*b*f=b^-1,c*d=d*c,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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