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## G = C6.S3≀C2order 432 = 24·33

### 4th non-split extension by C6 of S3≀C2 acting via S3≀C2/C32⋊C4=C2

Aliases: C6.4S3≀C2, He3⋊C41C4, He31(C4⋊C4), He3⋊C2.Q8, C2.1(He3⋊D4), (C2×He3).4D4, C3.(C3⋊S3.Q8), (C2×He3⋊C4).2C2, He3⋊(C2×C4).2C2, He3⋊C2.3(C2×C4), (C2×He3⋊C2).1C22, SmallGroup(432,237)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — He3 — He3⋊C2 — C6.S3≀C2
 Chief series C1 — C3 — He3 — He3⋊C2 — C2×He3⋊C2 — He3⋊(C2×C4) — C6.S3≀C2
 Lower central He3 — He3⋊C2 — C6.S3≀C2
 Upper central C1 — C2

Generators and relations for C6.S3≀C2
G = < a,b,c,d,e | a6=b3=c3=d4=1, e2=a3, ab=ba, ac=ca, ad=da, eae-1=a-1, cbc-1=a2b, dbd-1=a4c, ebe-1=b-1, dcd-1=a2b-1, ece-1=a2c, ede-1=d-1 >

Subgroups: 523 in 71 conjugacy classes, 15 normal (11 characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, C2×C4, C32, Dic3, C12, D6, C2×C6, C4⋊C4, C3×S3, C3×C6, C4×S3, C2×Dic3, C2×C12, He3, C3×Dic3, C3⋊Dic3, S3×C6, C4⋊Dic3, He3⋊C2, C2×He3, S3×Dic3, C32⋊C12, He3⋊C4, C2×He3⋊C2, He3⋊(C2×C4), C2×He3⋊C4, C6.S3≀C2
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C4⋊C4, S3≀C2, C3⋊S3.Q8, He3⋊D4, C6.S3≀C2

Character table of C6.S3≀C2

 class 1 2A 2B 2C 3A 3B 3C 4A 4B 4C 4D 4E 4F 6A 6B 6C 6D 6E 12A 12B 12C 12D 12E 12F 12G 12H size 1 1 9 9 2 12 12 18 18 18 18 18 18 2 12 12 18 18 18 18 18 18 36 36 36 36 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 1 1 1 1 -1 -1 -1 1 -1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ3 1 1 1 1 1 1 1 -1 1 -1 -1 -1 1 1 1 1 1 1 -1 -1 -1 -1 1 -1 1 -1 linear of order 2 ρ4 1 1 1 1 1 1 1 -1 -1 1 1 -1 -1 1 1 1 1 1 -1 -1 -1 -1 -1 1 -1 1 linear of order 2 ρ5 1 -1 -1 1 1 1 1 1 i -i i -1 -i -1 -1 -1 -1 1 1 -1 1 -1 i i -i -i linear of order 4 ρ6 1 -1 -1 1 1 1 1 1 -i i -i -1 i -1 -1 -1 -1 1 1 -1 1 -1 -i -i i i linear of order 4 ρ7 1 -1 -1 1 1 1 1 -1 i i -i 1 -i -1 -1 -1 -1 1 -1 1 -1 1 i -i -i i linear of order 4 ρ8 1 -1 -1 1 1 1 1 -1 -i -i i 1 i -1 -1 -1 -1 1 -1 1 -1 1 -i i i -i linear of order 4 ρ9 2 2 -2 -2 2 2 2 0 0 0 0 0 0 2 2 2 -2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 -2 2 -2 2 2 2 0 0 0 0 0 0 -2 -2 -2 2 -2 0 0 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ11 4 4 0 0 4 1 -2 0 0 -2 -2 0 0 4 1 -2 0 0 0 0 0 0 0 1 0 1 orthogonal lifted from S3≀C2 ρ12 4 4 0 0 4 1 -2 0 0 2 2 0 0 4 1 -2 0 0 0 0 0 0 0 -1 0 -1 orthogonal lifted from S3≀C2 ρ13 4 4 0 0 4 -2 1 0 -2 0 0 0 -2 4 -2 1 0 0 0 0 0 0 1 0 1 0 orthogonal lifted from S3≀C2 ρ14 4 4 0 0 4 -2 1 0 2 0 0 0 2 4 -2 1 0 0 0 0 0 0 -1 0 -1 0 orthogonal lifted from S3≀C2 ρ15 4 -4 0 0 4 -2 1 0 2i 0 0 0 -2i -4 2 -1 0 0 0 0 0 0 -i 0 i 0 complex lifted from C3⋊S3.Q8 ρ16 4 -4 0 0 4 1 -2 0 0 -2i 2i 0 0 -4 -1 2 0 0 0 0 0 0 0 -i 0 i complex lifted from C3⋊S3.Q8 ρ17 4 -4 0 0 4 1 -2 0 0 2i -2i 0 0 -4 -1 2 0 0 0 0 0 0 0 i 0 -i complex lifted from C3⋊S3.Q8 ρ18 4 -4 0 0 4 -2 1 0 -2i 0 0 0 2i -4 2 -1 0 0 0 0 0 0 i 0 -i 0 complex lifted from C3⋊S3.Q8 ρ19 6 6 -2 -2 -3 0 0 -2 0 0 0 -2 0 -3 0 0 1 1 1 1 1 1 0 0 0 0 orthogonal lifted from He3⋊D4 ρ20 6 6 2 2 -3 0 0 0 0 0 0 0 0 -3 0 0 -1 -1 -√3 √3 √3 -√3 0 0 0 0 orthogonal lifted from He3⋊D4 ρ21 6 6 -2 -2 -3 0 0 2 0 0 0 2 0 -3 0 0 1 1 -1 -1 -1 -1 0 0 0 0 orthogonal lifted from He3⋊D4 ρ22 6 6 2 2 -3 0 0 0 0 0 0 0 0 -3 0 0 -1 -1 √3 -√3 -√3 √3 0 0 0 0 orthogonal lifted from He3⋊D4 ρ23 6 -6 2 -2 -3 0 0 -2 0 0 0 2 0 3 0 0 -1 1 1 -1 1 -1 0 0 0 0 symplectic faithful, Schur index 2 ρ24 6 -6 -2 2 -3 0 0 0 0 0 0 0 0 3 0 0 1 -1 -√3 -√3 √3 √3 0 0 0 0 symplectic faithful, Schur index 2 ρ25 6 -6 -2 2 -3 0 0 0 0 0 0 0 0 3 0 0 1 -1 √3 √3 -√3 -√3 0 0 0 0 symplectic faithful, Schur index 2 ρ26 6 -6 2 -2 -3 0 0 2 0 0 0 -2 0 3 0 0 -1 1 -1 1 -1 1 0 0 0 0 symplectic faithful, Schur index 2

Smallest permutation representation of C6.S3≀C2
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 15 19)(2 16 20)(3 17 21)(4 18 22)(5 13 23)(6 14 24)(7 66 68)(8 61 69)(9 62 70)(10 63 71)(11 64 72)(12 65 67)(25 40 32)(26 41 33)(27 42 34)(28 37 35)(29 38 36)(30 39 31)(43 54 57)(44 49 58)(45 50 59)(46 51 60)(47 52 55)(48 53 56)
(1 13 19)(2 14 20)(3 15 21)(4 16 22)(5 17 23)(6 18 24)(7 72 64)(8 67 65)(9 68 66)(10 69 61)(11 70 62)(12 71 63)(25 42 34)(26 37 35)(27 38 36)(28 39 31)(29 40 32)(30 41 33)(43 59 54)(44 60 49)(45 55 50)(46 56 51)(47 57 52)(48 58 53)
(1 70 4 67)(2 71 5 68)(3 72 6 69)(7 22 63 15)(8 23 64 16)(9 24 65 17)(10 19 66 18)(11 20 61 13)(12 21 62 14)(25 57 37 46)(26 58 38 47)(27 59 39 48)(28 60 40 43)(29 55 41 44)(30 56 42 45)(31 49 34 52)(32 50 35 53)(33 51 36 54)
(1 50 4 53)(2 49 5 52)(3 54 6 51)(7 42 10 39)(8 41 11 38)(9 40 12 37)(13 47 16 44)(14 46 17 43)(15 45 18 48)(19 59 22 56)(20 58 23 55)(21 57 24 60)(25 65 28 62)(26 64 29 61)(27 63 30 66)(31 68 34 71)(32 67 35 70)(33 72 36 69)```

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

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

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

Matrix representation of C6.S3≀C2 in GL10(𝔽13)

 12 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 12 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 10 0 0 3 0 0 0 0 0 0 9 1 0 0 3 0 0 0 0 0 9 0 1 0 0 3
,
 0 12 0 0 0 0 0 0 0 0 1 12 0 0 0 0 0 0 0 0 0 9 0 1 0 0 0 0 0 0 4 0 12 12 0 0 0 0 0 0 0 0 0 0 3 0 6 0 0 0 0 0 0 0 3 0 10 0 0 0 0 0 0 0 12 1 10 0 0 0 0 0 0 0 9 0 1 1 0 2 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 1 12
,
 0 12 0 0 0 0 0 0 0 0 1 12 0 0 0 0 0 0 0 0 4 0 12 12 0 0 0 0 0 0 0 9 1 0 0 0 0 0 0 0 0 0 0 0 3 0 6 0 0 0 0 0 0 0 0 0 10 0 0 0 0 0 0 0 0 3 10 0 0 0 0 0 0 0 9 0 1 1 0 2 0 0 0 0 0 0 0 1 0 12 0 0 0 0 1 12 0 4 9 12
,
 0 0 5 8 0 0 0 0 0 0 6 6 10 5 0 0 0 0 0 0 1 1 7 0 0 0 0 0 0 0 1 9 7 0 0 0 0 0 0 0 0 0 0 0 8 11 11 0 0 0 0 0 0 0 6 0 5 0 0 0 0 0 0 0 6 5 0 0 0 0 0 0 0 0 6 10 10 8 11 11 0 0 0 0 2 7 0 6 5 0 0 0 0 0 2 0 7 6 0 5
,
 0 0 12 1 0 0 0 0 0 0 4 4 11 12 0 0 0 0 0 0 10 10 9 0 0 0 0 0 0 0 9 10 9 0 0 0 0 0 0 0 0 0 0 0 7 5 5 3 9 9 0 0 0 0 4 7 0 12 10 0 0 0 0 0 4 0 7 12 0 10 0 0 0 0 7 3 3 5 2 2 0 0 0 0 12 0 10 0 0 7 0 0 0 0 12 10 0 0 7 0

`G:=sub<GL(10,GF(13))| [12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,9,0,0,10,9,9,0,0,0,0,0,9,0,0,1,0,0,0,0,0,0,0,9,0,0,1,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,3],[0,1,0,4,0,0,0,0,0,0,12,12,9,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,0,0,3,3,12,9,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,6,10,10,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,2,12,12],[0,1,4,0,0,0,0,0,0,0,12,12,0,9,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,3,0,0,9,0,1,0,0,0,0,0,0,3,0,0,12,0,0,0,0,6,10,10,1,0,0,0,0,0,0,0,0,0,1,1,4,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,2,12,12],[0,6,1,1,0,0,0,0,0,0,0,6,1,9,0,0,0,0,0,0,5,10,7,7,0,0,0,0,0,0,8,5,0,0,0,0,0,0,0,0,0,0,0,0,8,6,6,6,2,2,0,0,0,0,11,0,5,10,7,0,0,0,0,0,11,5,0,10,0,7,0,0,0,0,0,0,0,8,6,6,0,0,0,0,0,0,0,11,5,0,0,0,0,0,0,0,0,11,0,5],[0,4,10,9,0,0,0,0,0,0,0,4,10,10,0,0,0,0,0,0,12,11,9,9,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,0,0,0,0,7,4,4,7,12,12,0,0,0,0,5,7,0,3,0,10,0,0,0,0,5,0,7,3,10,0,0,0,0,0,3,12,12,5,0,0,0,0,0,0,9,10,0,2,0,7,0,0,0,0,9,0,10,2,7,0] >;`

C6.S3≀C2 in GAP, Magma, Sage, TeX

`C_6.S_3\wr C_2`
`% in TeX`

`G:=Group("C6.S3wrC2");`
`// GroupNames label`

`G:=SmallGroup(432,237);`
`// by ID`

`G=gap.SmallGroup(432,237);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,56,85,64,1124,851,298,348,1027,537,14118,7069]);`
`// Polycyclic`

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

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