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G = C12×S4order 288 = 25·32

Direct product of C12 and S4

direct product, non-abelian, soluble, monomial

Aliases: C12×S4, (C2×S4).C6, A4⋊C42C6, (C4×A4)⋊2C6, C2.1(C6×S4), C22⋊(S3×C12), (C12×A4)⋊4C2, A41(C2×C12), (C6×S4).2C2, C6.40(C2×S4), (C22×C12)⋊1S3, C23.2(S3×C6), (C22×C6).9D6, (C6×A4).9C22, (C2×C6)⋊3(C4×S3), (C3×A4⋊C4)⋊5C2, (C3×A4)⋊4(C2×C4), (C2×A4).2(C2×C6), (C22×C4)⋊1(C3×S3), SmallGroup(288,897)

Series: Derived Chief Lower central Upper central

C1C22A4 — C12×S4
C1C22A4C2×A4C6×A4C6×S4 — C12×S4
A4 — C12×S4
C1C12

Generators and relations for C12×S4
 G = < a,b,c,d,e | a12=b2=c2=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=ebe=bc=cb, dcd-1=b, ce=ec, ede=d-1 >

Subgroups: 374 in 118 conjugacy classes, 28 normal (24 characteristic)
C1, C2, C2 [×4], C3, C3 [×2], C4, C4 [×5], C22, C22 [×6], S3 [×2], C6, C6 [×6], C2×C4 [×7], D4 [×4], C23, C23, C32, Dic3, C12, C12 [×7], A4, A4, D6, C2×C6, C2×C6 [×6], C42, C22⋊C4 [×2], C4⋊C4, C22×C4, C22×C4, C2×D4, C3×S3 [×2], C3×C6, C4×S3, C2×C12 [×7], C3×D4 [×4], S4 [×2], C2×A4, C2×A4, C22×C6, C22×C6, C4×D4, C3×Dic3, C3×C12, C3×A4, S3×C6, C4×C12, C3×C22⋊C4 [×2], C3×C4⋊C4, A4⋊C4, C4×A4, C4×A4, C22×C12, C22×C12, C6×D4, C2×S4, S3×C12, C3×S4 [×2], C6×A4, D4×C12, C4×S4, C3×A4⋊C4, C12×A4, C6×S4, C12×S4
Quotients: C1, C2 [×3], C3, C4 [×2], C22, S3, C6 [×3], C2×C4, C12 [×2], D6, C2×C6, C3×S3, C4×S3, C2×C12, S4, S3×C6, C2×S4, S3×C12, C3×S4, C4×S4, C6×S4, C12×S4

Smallest permutation representation of C12×S4
On 36 points
Generators in S36
(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 7)(2 8)(3 9)(4 10)(5 11)(6 12)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)
(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)
(1 34 20)(2 35 21)(3 36 22)(4 25 23)(5 26 24)(6 27 13)(7 28 14)(8 29 15)(9 30 16)(10 31 17)(11 32 18)(12 33 19)
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 33)(14 34)(15 35)(16 36)(17 25)(18 26)(19 27)(20 28)(21 29)(22 30)(23 31)(24 32)

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

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

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

60 conjugacy classes

class 1 2A2B2C2D2E3A3B3C3D3E4A4B4C4D4E···4J6A6B6C6D6E6F6G6H6I6J6K6L6M12A12B12C12D12E12F12G12H12I···12T12U···12Z
order1222223333344444···46666666666666121212121212121212···1212···12
size1133661188811336···61133336666888111133336···68···8

60 irreducible representations

dim1111111111222222333333
type++++++++
imageC1C2C2C2C3C4C6C6C6C12S3D6C3×S3C4×S3S3×C6S3×C12S4C2×S4C3×S4C4×S4C6×S4C12×S4
kernelC12×S4C3×A4⋊C4C12×A4C6×S4C4×S4C3×S4A4⋊C4C4×A4C2×S4S4C22×C12C22×C6C22×C4C2×C6C23C22C12C6C4C3C2C1
# reps1111242228112224224448

Matrix representation of C12×S4 in GL3(𝔽13) generated by

200
020
002
,
1200
0120
001
,
100
0120
0012
,
001
100
010
,
1200
0012
0120
G:=sub<GL(3,GF(13))| [2,0,0,0,2,0,0,0,2],[12,0,0,0,12,0,0,0,1],[1,0,0,0,12,0,0,0,12],[0,1,0,0,0,1,1,0,0],[12,0,0,0,0,12,0,12,0] >;

C12×S4 in GAP, Magma, Sage, TeX

C_{12}\times S_4
% in TeX

G:=Group("C12xS4");
// GroupNames label

G:=SmallGroup(288,897);
// by ID

G=gap.SmallGroup(288,897);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-3,-2,2,92,1684,6053,285,3534,475]);
// Polycyclic

G:=Group<a,b,c,d,e|a^12=b^2=c^2=d^3=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,d*b*d^-1=e*b*e=b*c=c*b,d*c*d^-1=b,c*e=e*c,e*d*e=d^-1>;
// generators/relations

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