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## G = (C2×C6)⋊4S4order 288 = 25·32

### 2nd semidirect product of C2×C6 and S4 acting via S4/A4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C6×A4 — (C2×C6)⋊4S4
 Chief series C1 — C22 — C2×C6 — C3×A4 — C6×A4 — C2×C3⋊S4 — (C2×C6)⋊4S4
 Lower central C3×A4 — C6×A4 — (C2×C6)⋊4S4
 Upper central C1 — C2 — C22

Generators and relations for (C2×C6)⋊4S4
G = < a,b,c,d,e,f | a2=b6=c2=d2=e3=f2=1, ab=ba, ac=ca, ad=da, ae=ea, faf=ab3, bc=cb, bd=db, be=eb, fbf=b-1, ece-1=fcf=cd=dc, ede-1=c, df=fd, fef=e-1 >

Subgroups: 952 in 156 conjugacy classes, 27 normal (18 characteristic)
C1, C2, C2 [×5], C3, C3 [×3], C4 [×3], C22 [×2], C22 [×11], S3 [×4], C6, C6 [×10], C2×C4 [×3], D4 [×6], C23, C23 [×5], C32, Dic3 [×6], A4 [×3], D6 [×6], C2×C6 [×2], C2×C6 [×11], C22⋊C4 [×3], C2×D4 [×3], C24, C3⋊S3, C3×C6 [×2], C2×Dic3 [×3], C3⋊D4 [×9], S4 [×3], C2×A4 [×3], C2×A4 [×3], C22×S3, C22×C6, C22×C6 [×4], C22≀C2, C3⋊Dic3, C3×A4, C2×C3⋊S3, C62, C6.D4 [×3], A4⋊C4 [×3], C2×C3⋊D4 [×3], C2×S4 [×3], C22×A4 [×3], C23×C6, C327D4, C3⋊S4, C6×A4, C6×A4, C244S3, A4⋊D4 [×3], C6.7S4, C2×C3⋊S4, A4×C2×C6, (C2×C6)⋊4S4
Quotients: C1, C2 [×3], C22, S3 [×4], D4, D6 [×4], C3⋊S3, C3⋊D4 [×4], S4, C2×C3⋊S3, C2×S4, C327D4, C3⋊S4, A4⋊D4, C2×C3⋊S4, (C2×C6)⋊4S4

Character table of (C2×C6)⋊4S4

 class 1 2A 2B 2C 2D 2E 2F 3A 3B 3C 3D 4A 4B 4C 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J 6K 6L 6M 6N 6O 6P size 1 1 2 3 3 6 36 2 8 8 8 36 36 36 2 2 2 6 6 6 6 8 8 8 8 8 8 8 8 8 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 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 -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 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 -1 1 1 -1 linear of order 2 ρ5 2 2 -2 2 2 -2 0 -1 -1 2 -1 0 0 0 1 -1 1 -1 1 1 -1 -1 1 1 1 -2 1 -1 2 -2 orthogonal lifted from D6 ρ6 2 2 2 2 2 2 0 -1 2 -1 -1 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 2 -1 -1 2 2 -1 -1 orthogonal lifted from S3 ρ7 2 2 -2 2 2 -2 0 2 -1 -1 -1 0 0 0 -2 2 -2 2 -2 -2 2 -1 1 1 1 1 1 -1 -1 1 orthogonal lifted from D6 ρ8 2 2 -2 2 2 -2 0 -1 2 -1 -1 0 0 0 1 -1 1 -1 1 1 -1 -1 1 -2 1 1 -2 2 -1 1 orthogonal lifted from D6 ρ9 2 2 2 2 2 2 0 2 -1 -1 -1 0 0 0 2 2 2 2 2 2 2 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ10 2 -2 0 -2 2 0 0 2 2 2 2 0 0 0 0 -2 0 2 0 0 -2 -2 0 0 0 0 0 -2 -2 0 orthogonal lifted from D4 ρ11 2 2 2 2 2 2 0 -1 -1 -1 2 0 0 0 -1 -1 -1 -1 -1 -1 -1 2 2 -1 2 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ12 2 2 2 2 2 2 0 -1 -1 2 -1 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 2 -1 -1 2 2 orthogonal lifted from S3 ρ13 2 2 -2 2 2 -2 0 -1 -1 -1 2 0 0 0 1 -1 1 -1 1 1 -1 2 -2 1 -2 1 1 -1 -1 1 orthogonal lifted from D6 ρ14 2 -2 0 -2 2 0 0 -1 -1 2 -1 0 0 0 -√-3 1 √-3 -1 √-3 -√-3 1 1 √-3 √-3 -√-3 0 -√-3 1 -2 0 complex lifted from C3⋊D4 ρ15 2 -2 0 -2 2 0 0 -1 2 -1 -1 0 0 0 √-3 1 -√-3 -1 -√-3 √-3 1 1 √-3 0 -√-3 √-3 0 -2 1 -√-3 complex lifted from C3⋊D4 ρ16 2 -2 0 -2 2 0 0 2 -1 -1 -1 0 0 0 0 -2 0 2 0 0 -2 1 -√-3 √-3 √-3 √-3 -√-3 1 1 -√-3 complex lifted from C3⋊D4 ρ17 2 -2 0 -2 2 0 0 -1 2 -1 -1 0 0 0 -√-3 1 √-3 -1 √-3 -√-3 1 1 -√-3 0 √-3 -√-3 0 -2 1 √-3 complex lifted from C3⋊D4 ρ18 2 -2 0 -2 2 0 0 -1 -1 -1 2 0 0 0 √-3 1 -√-3 -1 -√-3 √-3 1 -2 0 √-3 0 -√-3 -√-3 1 1 √-3 complex lifted from C3⋊D4 ρ19 2 -2 0 -2 2 0 0 -1 -1 -1 2 0 0 0 -√-3 1 √-3 -1 √-3 -√-3 1 -2 0 -√-3 0 √-3 √-3 1 1 -√-3 complex lifted from C3⋊D4 ρ20 2 -2 0 -2 2 0 0 -1 -1 2 -1 0 0 0 √-3 1 -√-3 -1 -√-3 √-3 1 1 -√-3 -√-3 √-3 0 √-3 1 -2 0 complex lifted from C3⋊D4 ρ21 2 -2 0 -2 2 0 0 2 -1 -1 -1 0 0 0 0 -2 0 2 0 0 -2 1 √-3 -√-3 -√-3 -√-3 √-3 1 1 √-3 complex lifted from C3⋊D4 ρ22 3 3 3 -1 -1 -1 1 3 0 0 0 -1 1 -1 3 3 3 -1 -1 -1 -1 0 0 0 0 0 0 0 0 0 orthogonal lifted from S4 ρ23 3 3 -3 -1 -1 1 1 3 0 0 0 1 -1 -1 -3 3 -3 -1 1 1 -1 0 0 0 0 0 0 0 0 0 orthogonal lifted from C2×S4 ρ24 3 3 3 -1 -1 -1 -1 3 0 0 0 1 -1 1 3 3 3 -1 -1 -1 -1 0 0 0 0 0 0 0 0 0 orthogonal lifted from S4 ρ25 3 3 -3 -1 -1 1 -1 3 0 0 0 -1 1 1 -3 3 -3 -1 1 1 -1 0 0 0 0 0 0 0 0 0 orthogonal lifted from C2×S4 ρ26 6 6 6 -2 -2 -2 0 -3 0 0 0 0 0 0 -3 -3 -3 1 1 1 1 0 0 0 0 0 0 0 0 0 orthogonal lifted from C3⋊S4 ρ27 6 6 -6 -2 -2 2 0 -3 0 0 0 0 0 0 3 -3 3 1 -1 -1 1 0 0 0 0 0 0 0 0 0 orthogonal lifted from C2×C3⋊S4 ρ28 6 -6 0 2 -2 0 0 6 0 0 0 0 0 0 0 -6 0 -2 0 0 2 0 0 0 0 0 0 0 0 0 orthogonal lifted from A4⋊D4 ρ29 6 -6 0 2 -2 0 0 -3 0 0 0 0 0 0 3√-3 3 -3√-3 1 √-3 -√-3 -1 0 0 0 0 0 0 0 0 0 complex faithful ρ30 6 -6 0 2 -2 0 0 -3 0 0 0 0 0 0 -3√-3 3 3√-3 1 -√-3 √-3 -1 0 0 0 0 0 0 0 0 0 complex faithful

Smallest permutation representation of (C2×C6)⋊4S4
On 36 points
Generators in S36
(1 18)(2 13)(3 14)(4 15)(5 16)(6 17)(7 36)(8 31)(9 32)(10 33)(11 34)(12 35)(19 25)(20 26)(21 27)(22 28)(23 29)(24 30)
(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)
(1 15)(2 16)(3 17)(4 18)(5 13)(6 14)(7 10)(8 11)(9 12)(19 25)(20 26)(21 27)(22 28)(23 29)(24 30)(31 34)(32 35)(33 36)
(1 4)(2 5)(3 6)(7 36)(8 31)(9 32)(10 33)(11 34)(12 35)(13 16)(14 17)(15 18)(19 28)(20 29)(21 30)(22 25)(23 26)(24 27)
(1 26 9)(2 27 10)(3 28 11)(4 29 12)(5 30 7)(6 25 8)(13 21 33)(14 22 34)(15 23 35)(16 24 36)(17 19 31)(18 20 32)
(2 6)(3 5)(7 28)(8 27)(9 26)(10 25)(11 30)(12 29)(13 14)(15 18)(16 17)(19 36)(20 35)(21 34)(22 33)(23 32)(24 31)

G:=sub<Sym(36)| (1,18)(2,13)(3,14)(4,15)(5,16)(6,17)(7,36)(8,31)(9,32)(10,33)(11,34)(12,35)(19,25)(20,26)(21,27)(22,28)(23,29)(24,30), (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), (1,15)(2,16)(3,17)(4,18)(5,13)(6,14)(7,10)(8,11)(9,12)(19,25)(20,26)(21,27)(22,28)(23,29)(24,30)(31,34)(32,35)(33,36), (1,4)(2,5)(3,6)(7,36)(8,31)(9,32)(10,33)(11,34)(12,35)(13,16)(14,17)(15,18)(19,28)(20,29)(21,30)(22,25)(23,26)(24,27), (1,26,9)(2,27,10)(3,28,11)(4,29,12)(5,30,7)(6,25,8)(13,21,33)(14,22,34)(15,23,35)(16,24,36)(17,19,31)(18,20,32), (2,6)(3,5)(7,28)(8,27)(9,26)(10,25)(11,30)(12,29)(13,14)(15,18)(16,17)(19,36)(20,35)(21,34)(22,33)(23,32)(24,31)>;

G:=Group( (1,18)(2,13)(3,14)(4,15)(5,16)(6,17)(7,36)(8,31)(9,32)(10,33)(11,34)(12,35)(19,25)(20,26)(21,27)(22,28)(23,29)(24,30), (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), (1,15)(2,16)(3,17)(4,18)(5,13)(6,14)(7,10)(8,11)(9,12)(19,25)(20,26)(21,27)(22,28)(23,29)(24,30)(31,34)(32,35)(33,36), (1,4)(2,5)(3,6)(7,36)(8,31)(9,32)(10,33)(11,34)(12,35)(13,16)(14,17)(15,18)(19,28)(20,29)(21,30)(22,25)(23,26)(24,27), (1,26,9)(2,27,10)(3,28,11)(4,29,12)(5,30,7)(6,25,8)(13,21,33)(14,22,34)(15,23,35)(16,24,36)(17,19,31)(18,20,32), (2,6)(3,5)(7,28)(8,27)(9,26)(10,25)(11,30)(12,29)(13,14)(15,18)(16,17)(19,36)(20,35)(21,34)(22,33)(23,32)(24,31) );

G=PermutationGroup([(1,18),(2,13),(3,14),(4,15),(5,16),(6,17),(7,36),(8,31),(9,32),(10,33),(11,34),(12,35),(19,25),(20,26),(21,27),(22,28),(23,29),(24,30)], [(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)], [(1,15),(2,16),(3,17),(4,18),(5,13),(6,14),(7,10),(8,11),(9,12),(19,25),(20,26),(21,27),(22,28),(23,29),(24,30),(31,34),(32,35),(33,36)], [(1,4),(2,5),(3,6),(7,36),(8,31),(9,32),(10,33),(11,34),(12,35),(13,16),(14,17),(15,18),(19,28),(20,29),(21,30),(22,25),(23,26),(24,27)], [(1,26,9),(2,27,10),(3,28,11),(4,29,12),(5,30,7),(6,25,8),(13,21,33),(14,22,34),(15,23,35),(16,24,36),(17,19,31),(18,20,32)], [(2,6),(3,5),(7,28),(8,27),(9,26),(10,25),(11,30),(12,29),(13,14),(15,18),(16,17),(19,36),(20,35),(21,34),(22,33),(23,32),(24,31)])

Matrix representation of (C2×C6)⋊4S4 in GL5(𝔽13)

 1 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 12
,
 4 0 0 0 0 0 10 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 12 0 0 0 0 0 12
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 0
,
 0 1 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0

G:=sub<GL(5,GF(13))| [1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[4,0,0,0,0,0,10,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0],[0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0] >;

(C2×C6)⋊4S4 in GAP, Magma, Sage, TeX

(C_2\times C_6)\rtimes_4S_4
% in TeX

G:=Group("(C2xC6):4S4");
// GroupNames label

G:=SmallGroup(288,917);
// by ID

G=gap.SmallGroup(288,917);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-3,-2,2,85,451,1684,6053,782,3534,1350]);
// Polycyclic

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

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