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## G = C6.(S3×A4)  order 432 = 24·33

### 7th non-split extension by C6 of S3×A4 acting via S3×A4/C3×A4=C2

Aliases: C6.7(S3×A4), Q8⋊He33C2, C3⋊Dic3.A4, C32⋊(C4.A4), (Q8×C32).4C6, C12.26D61C3, Q8.2(C32⋊C6), (C3×SL2(𝔽3))⋊1S3, C2.2(C62⋊C6), C3.3(Dic3.A4), (C3×C6).1(C2×A4), (C3×Q8).15(C3×S3), SmallGroup(432,269)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8×C32 — C6.(S3×A4)
 Chief series C1 — C2 — C6 — C3×C6 — Q8×C32 — Q8⋊He3 — C6.(S3×A4)
 Lower central Q8×C32 — C6.(S3×A4)
 Upper central C1 — C2

Generators and relations for C6.(S3×A4)
G = < a,b,c,d,e,f | a6=b3=f3=1, c2=d2=e2=a3, ab=ba, cac-1=a-1, ad=da, ae=ea, af=fa, cbc-1=b-1, bd=db, be=eb, fbf-1=a2b, cd=dc, ce=ec, cf=fc, ede-1=fef-1=a3d, fdf-1=de >

Subgroups: 592 in 71 conjugacy classes, 14 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, C2×C4, D4, Q8, C32, C32, Dic3, C12, D6, C4○D4, C3⋊S3, C3×C6, C3×C6, SL2(𝔽3), C4×S3, D12, C3×Q8, C3×Q8, He3, C3×Dic3, C3⋊Dic3, C3×C12, C2×C3⋊S3, C4.A4, Q83S3, C2×He3, C3×SL2(𝔽3), C3×SL2(𝔽3), C4×C3⋊S3, C12⋊S3, Q8×C32, C32⋊C12, Dic3.A4, C12.26D6, Q8⋊He3, C6.(S3×A4)
Quotients: C1, C2, C3, S3, C6, A4, C3×S3, C2×A4, C4.A4, C32⋊C6, S3×A4, Dic3.A4, C62⋊C6, C6.(S3×A4)

Character table of C6.(S3×A4)

 class 1 2A 2B 3A 3B 3C 3D 3E 3F 4A 4B 4C 6A 6B 6C 6D 6E 6F 12A 12B 12C 12D 12E 12F 12G 12H size 1 1 54 2 6 12 12 24 24 6 9 9 2 6 12 12 24 24 12 12 12 12 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 ζ32 ζ3 ζ32 ζ3 1 1 1 1 1 ζ32 ζ3 ζ3 ζ32 1 1 1 1 ζ32 ζ32 ζ3 ζ3 linear of order 3 ρ4 1 1 -1 1 1 ζ32 ζ3 ζ32 ζ3 1 -1 -1 1 1 ζ32 ζ3 ζ3 ζ32 1 1 1 1 ζ6 ζ6 ζ65 ζ65 linear of order 6 ρ5 1 1 1 1 1 ζ3 ζ32 ζ3 ζ32 1 1 1 1 1 ζ3 ζ32 ζ32 ζ3 1 1 1 1 ζ3 ζ3 ζ32 ζ32 linear of order 3 ρ6 1 1 -1 1 1 ζ3 ζ32 ζ3 ζ32 1 -1 -1 1 1 ζ3 ζ32 ζ32 ζ3 1 1 1 1 ζ65 ζ65 ζ6 ζ6 linear of order 6 ρ7 2 2 0 2 -1 2 2 -1 -1 2 0 0 2 -1 2 2 -1 -1 -1 2 -1 -1 0 0 0 0 orthogonal lifted from S3 ρ8 2 2 0 2 -1 -1+√-3 -1-√-3 ζ65 ζ6 2 0 0 2 -1 -1+√-3 -1-√-3 ζ6 ζ65 -1 2 -1 -1 0 0 0 0 complex lifted from C3×S3 ρ9 2 2 0 2 -1 -1-√-3 -1+√-3 ζ6 ζ65 2 0 0 2 -1 -1-√-3 -1+√-3 ζ65 ζ6 -1 2 -1 -1 0 0 0 0 complex lifted from C3×S3 ρ10 2 -2 0 2 2 -1 -1 -1 -1 0 -2i 2i -2 -2 1 1 1 1 0 0 0 0 -i i -i i complex lifted from C4.A4 ρ11 2 -2 0 2 2 -1 -1 -1 -1 0 2i -2i -2 -2 1 1 1 1 0 0 0 0 i -i i -i complex lifted from C4.A4 ρ12 2 -2 0 2 2 ζ6 ζ65 ζ6 ζ65 0 -2i 2i -2 -2 ζ32 ζ3 ζ3 ζ32 0 0 0 0 ζ43ζ32 ζ4ζ32 ζ43ζ3 ζ4ζ3 complex lifted from C4.A4 ρ13 2 -2 0 2 2 ζ65 ζ6 ζ65 ζ6 0 2i -2i -2 -2 ζ3 ζ32 ζ32 ζ3 0 0 0 0 ζ4ζ3 ζ43ζ3 ζ4ζ32 ζ43ζ32 complex lifted from C4.A4 ρ14 2 -2 0 2 2 ζ65 ζ6 ζ65 ζ6 0 -2i 2i -2 -2 ζ3 ζ32 ζ32 ζ3 0 0 0 0 ζ43ζ3 ζ4ζ3 ζ43ζ32 ζ4ζ32 complex lifted from C4.A4 ρ15 2 -2 0 2 2 ζ6 ζ65 ζ6 ζ65 0 2i -2i -2 -2 ζ32 ζ3 ζ3 ζ32 0 0 0 0 ζ4ζ32 ζ43ζ32 ζ4ζ3 ζ43ζ3 complex lifted from C4.A4 ρ16 3 3 1 3 3 0 0 0 0 -1 -3 -3 3 3 0 0 0 0 -1 -1 -1 -1 0 0 0 0 orthogonal lifted from C2×A4 ρ17 3 3 -1 3 3 0 0 0 0 -1 3 3 3 3 0 0 0 0 -1 -1 -1 -1 0 0 0 0 orthogonal lifted from A4 ρ18 4 -4 0 4 -2 -2 -2 1 1 0 0 0 -4 2 2 2 -1 -1 0 0 0 0 0 0 0 0 orthogonal lifted from Dic3.A4, Schur index 2 ρ19 4 -4 0 4 -2 1-√-3 1+√-3 ζ3 ζ32 0 0 0 -4 2 -1+√-3 -1-√-3 ζ6 ζ65 0 0 0 0 0 0 0 0 complex lifted from Dic3.A4 ρ20 4 -4 0 4 -2 1+√-3 1-√-3 ζ32 ζ3 0 0 0 -4 2 -1-√-3 -1+√-3 ζ65 ζ6 0 0 0 0 0 0 0 0 complex lifted from Dic3.A4 ρ21 6 6 0 6 -3 0 0 0 0 -2 0 0 6 -3 0 0 0 0 1 -2 1 1 0 0 0 0 orthogonal lifted from S3×A4 ρ22 6 6 0 -3 0 0 0 0 0 -2 0 0 -3 0 0 0 0 0 -2 1 4 -2 0 0 0 0 orthogonal lifted from C62⋊C6 ρ23 6 6 0 -3 0 0 0 0 0 -2 0 0 -3 0 0 0 0 0 -2 1 -2 4 0 0 0 0 orthogonal lifted from C62⋊C6 ρ24 6 6 0 -3 0 0 0 0 0 -2 0 0 -3 0 0 0 0 0 4 1 -2 -2 0 0 0 0 orthogonal lifted from C62⋊C6 ρ25 6 6 0 -3 0 0 0 0 0 6 0 0 -3 0 0 0 0 0 0 -3 0 0 0 0 0 0 orthogonal lifted from C32⋊C6 ρ26 12 -12 0 -6 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal faithful, Schur index 2

Smallest permutation representation of C6.(S3×A4)
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)
(25 27 29)(26 28 30)(31 33 35)(32 34 36)(37 39 41)(38 40 42)(43 45 47)(44 46 48)(49 53 51)(50 54 52)(55 59 57)(56 60 58)(61 65 63)(62 66 64)(67 71 69)(68 72 70)
(1 16 4 13)(2 15 5 18)(3 14 6 17)(7 19 10 22)(8 24 11 21)(9 23 12 20)(25 40 28 37)(26 39 29 42)(27 38 30 41)(31 43 34 46)(32 48 35 45)(33 47 36 44)(49 64 52 61)(50 63 53 66)(51 62 54 65)(55 67 58 70)(56 72 59 69)(57 71 60 68)
(1 13 4 16)(2 14 5 17)(3 15 6 18)(7 19 10 22)(8 20 11 23)(9 21 12 24)(25 43 28 46)(26 44 29 47)(27 45 30 48)(31 40 34 37)(32 41 35 38)(33 42 36 39)(49 55 52 58)(50 56 53 59)(51 57 54 60)(61 70 64 67)(62 71 65 68)(63 72 66 69)
(1 10 4 7)(2 11 5 8)(3 12 6 9)(13 19 16 22)(14 20 17 23)(15 21 18 24)(25 40 28 37)(26 41 29 38)(27 42 30 39)(31 46 34 43)(32 47 35 44)(33 48 36 45)(49 70 52 67)(50 71 53 68)(51 72 54 69)(55 61 58 64)(56 62 59 65)(57 63 60 66)
(1 25 49)(2 26 50)(3 27 51)(4 28 52)(5 29 53)(6 30 54)(7 31 55)(8 32 56)(9 33 57)(10 34 58)(11 35 59)(12 36 60)(13 37 61)(14 38 62)(15 39 63)(16 40 64)(17 41 65)(18 42 66)(19 43 67)(20 44 68)(21 45 69)(22 46 70)(23 47 71)(24 48 72)

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

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

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

Matrix representation of C6.(S3×A4) 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 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 0 12 0 0 0 0 0 0 0 0 1 12
,
 12 12 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 12 12 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 0 0 0 1 0 0 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 0 12 0 0 0 0 0 0 0 0 1 12
,
 8 0 0 0 0 0 0 0 0 0 5 5 0 0 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 0 5 5 0 0 0 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 0 0 0 1 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 0 0 0 1 0
,
 5 0 0 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 0 0 8 0 0 0 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 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 12
,
 0 0 1 0 0 0 0 0 0 0 0 0 0 1 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 12 0 0 0 0 0 0 0 0 0 0 12 0 0 0 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 0 0 0 12 0 0 0 0 0 0 0 0 0 0 12
,
 12 0 1 0 0 0 0 0 0 0 0 12 0 1 0 0 0 0 0 0 5 0 5 0 0 0 0 0 0 0 0 5 0 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 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 0 0 0 1 0 0

G:=sub<GL(10,GF(13))| [12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,12],[12,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,12],[8,5,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,8,5,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0],[5,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12],[0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12],[12,0,5,0,0,0,0,0,0,0,0,12,0,5,0,0,0,0,0,0,1,0,5,0,0,0,0,0,0,0,0,1,0,5,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0] >;

C6.(S3×A4) in GAP, Magma, Sage, TeX

C_6.(S_3\times A_4)
% in TeX

G:=Group("C6.(S3xA4)");
// GroupNames label

G:=SmallGroup(432,269);
// by ID

G=gap.SmallGroup(432,269);
# by ID

G:=PCGroup([7,-2,-3,-2,2,-3,-3,-2,1512,198,772,94,1081,528,1684,1691,6053]);
// Polycyclic

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

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