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## G = C12.73S32order 432 = 24·33

### 30th 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 — C12.73S32
 Chief series C1 — C3 — C32 — C33 — C32×C6 — S3×C3×C6 — C33⋊6D4 — C12.73S32
 Lower central C33 — C32×C6 — C12.73S32
 Upper central C1 — C4

Generators and relations for C12.73S32
G = < a,b,c,d,e | a12=b3=c2=d3=e2=1, ab=ba, ac=ca, ad=da, eae=a5, cbc=b-1, bd=db, be=eb, cd=dc, ece=a6c, ede=d-1 >

Subgroups: 1784 in 304 conjugacy classes, 68 normal (32 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, C33, C3×Dic3, C3×Dic3, C3⋊Dic3, C3⋊Dic3, C3×C12, C3×C12, C3×C12, S3×C6, S3×C6, C2×C3⋊S3, C2×C3⋊S3, C62, C4○D12, S3×C32, C3×C3⋊S3, C33⋊C2, C32×C6, D6⋊S3, C3⋊D12, C322Q8, S3×C12, S3×C12, C324Q8, C4×C3⋊S3, C4×C3⋊S3, C12⋊S3, C327D4, C6×C12, C32×Dic3, C3×C3⋊Dic3, C335C4, C32×C12, S3×C3×C6, C6×C3⋊S3, C2×C33⋊C2, D6.D6, C12.59D6, C336D4, C337D4, C338D4, C334Q8, S3×C3×C12, C12×C3⋊S3, C4×C33⋊C2, C12.73S32
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C3⋊S3, C22×S3, S32, C2×C3⋊S3, C4○D12, C2×S32, C22×C3⋊S3, S3×C3⋊S3, D6.D6, C12.59D6, C2×S3×C3⋊S3, C12.73S32

Smallest permutation representation of C12.73S32
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 37 17)(2 38 18)(3 39 19)(4 40 20)(5 41 21)(6 42 22)(7 43 23)(8 44 24)(9 45 13)(10 46 14)(11 47 15)(12 48 16)(25 67 49)(26 68 50)(27 69 51)(28 70 52)(29 71 53)(30 72 54)(31 61 55)(32 62 56)(33 63 57)(34 64 58)(35 65 59)(36 66 60)
(1 28)(2 29)(3 30)(4 31)(5 32)(6 33)(7 34)(8 35)(9 36)(10 25)(11 26)(12 27)(13 66)(14 67)(15 68)(16 69)(17 70)(18 71)(19 72)(20 61)(21 62)(22 63)(23 64)(24 65)(37 52)(38 53)(39 54)(40 55)(41 56)(42 57)(43 58)(44 59)(45 60)(46 49)(47 50)(48 51)
(1 21 45)(2 22 46)(3 23 47)(4 24 48)(5 13 37)(6 14 38)(7 15 39)(8 16 40)(9 17 41)(10 18 42)(11 19 43)(12 20 44)(25 71 57)(26 72 58)(27 61 59)(28 62 60)(29 63 49)(30 64 50)(31 65 51)(32 66 52)(33 67 53)(34 68 54)(35 69 55)(36 70 56)
(1 25)(2 30)(3 35)(4 28)(5 33)(6 26)(7 31)(8 36)(9 29)(10 34)(11 27)(12 32)(13 53)(14 58)(15 51)(16 56)(17 49)(18 54)(19 59)(20 52)(21 57)(22 50)(23 55)(24 60)(37 67)(38 72)(39 65)(40 70)(41 63)(42 68)(43 61)(44 66)(45 71)(46 64)(47 69)(48 62)```

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

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

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

66 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 6R 6S 12A ··· 12J 12K ··· 12R 12S ··· 12Z 12AA 12AB 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 6 6 12 ··· 12 12 ··· 12 12 ··· 12 12 12 size 1 1 6 18 54 2 ··· 2 4 4 4 4 1 1 6 18 54 2 ··· 2 4 4 4 4 6 ··· 6 18 18 2 ··· 2 4 ··· 4 6 ··· 6 18 18

66 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 S3 S3 D6 D6 D6 D6 D6 C4○D4 C4○D12 S32 C2×S32 D6.D6 kernel C12.73S32 C33⋊6D4 C33⋊7D4 C33⋊8D4 C33⋊4Q8 S3×C3×C12 C12×C3⋊S3 C4×C33⋊C2 S3×C12 C4×C3⋊S3 C3×Dic3 C3⋊Dic3 C3×C12 S3×C6 C2×C3⋊S3 C33 C32 C12 C6 C3 # reps 1 1 1 1 1 1 1 1 4 1 4 1 5 4 1 2 20 4 4 8

Matrix representation of C12.73S32 in GL8(𝔽13)

 5 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 1 12 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 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 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
,
 7 2 0 0 0 0 0 0 2 6 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 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
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 12 1 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 12 1 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
,
 4 3 0 0 0 0 0 0 8 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 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1

`G:=sub<GL(8,GF(13))| [5,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12],[7,2,0,0,0,0,0,0,2,6,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[4,8,0,0,0,0,0,0,3,9,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1] >;`

C12.73S32 in GAP, Magma, Sage, TeX

`C_{12}._{73}S_3^2`
`% in TeX`

`G:=Group("C12.73S3^2");`
`// GroupNames label`

`G:=SmallGroup(432,667);`
`// by ID`

`G=gap.SmallGroup(432,667);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,141,58,571,2028,14118]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^12=b^3=c^2=d^3=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e=a^5,c*b*c=b^-1,b*d=d*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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