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G = C12⋊S4order 288 = 25·32

1st semidirect product of C12 and S4 acting via S4/A4=C2

non-abelian, soluble, monomial

Aliases: C121S4, A41D12, C4⋊(C3⋊S4), C31(C4⋊S4), (C4×A4)⋊1S3, (C3×A4)⋊5D4, (C2×C6)⋊3D12, (C12×A4)⋊1C2, C6.30(C2×S4), (C22×C12)⋊2S3, (C2×A4).10D6, C22⋊(C12⋊S3), (C22×C6).21D6, (C6×A4).15C22, (C2×C3⋊S4)⋊3C2, C2.4(C2×C3⋊S4), C23.3(C2×C3⋊S3), (C22×C4)⋊2(C3⋊S3), SmallGroup(288,909)

Series: Derived Chief Lower central Upper central

C1C22C6×A4 — C12⋊S4
C1C22C2×C6C3×A4C6×A4C2×C3⋊S4 — C12⋊S4
C3×A4C6×A4 — C12⋊S4
C1C2C4

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

Subgroups: 1028 in 144 conjugacy classes, 27 normal (16 characteristic)
C1, C2, C2 [×4], C3, C3 [×3], C4, C4 [×3], C22, C22 [×8], S3 [×8], C6, C6 [×5], C2×C4 [×4], D4 [×6], C23, C23 [×2], C32, Dic3 [×2], C12, C12 [×4], A4 [×3], D6 [×12], C2×C6, C2×C6 [×2], C22⋊C4 [×2], C4⋊C4, C22×C4, C2×D4 [×3], C3⋊S3 [×2], C3×C6, D12 [×5], C2×Dic3 [×2], C3⋊D4 [×4], C2×C12 [×2], S4 [×6], C2×A4 [×3], C22×S3 [×2], C22×C6, C4⋊D4, C3×C12, C3×A4, C2×C3⋊S3 [×2], C4⋊Dic3, D6⋊C4 [×2], C4×A4 [×3], C2×D12, C2×C3⋊D4 [×2], C22×C12, C2×S4 [×6], C12⋊S3, C3⋊S4 [×2], C6×A4, C127D4, C4⋊S4 [×3], C12×A4, C2×C3⋊S4 [×2], C12⋊S4
Quotients: C1, C2 [×3], C22, S3 [×4], D4, D6 [×4], C3⋊S3, D12 [×4], S4, C2×C3⋊S3, C2×S4, C12⋊S3, C3⋊S4, C4⋊S4, C2×C3⋊S4, C12⋊S4

Character table of C12⋊S4

 class 12A2B2C2D2E3A3B3C3D4A4B4C4D6A6B6C6D6E6F12A12B12C12D12E12F12G12H12I12J
 size 1133363628882636362668882266888888
ρ1111111111111111111111111111111    trivial
ρ21111-1-1111111-1-11111111111111111    linear of order 2
ρ31111-111111-1-1-11111111-1-1-1-1-1-1-1-1-1-1    linear of order 2
ρ411111-11111-1-11-1111111-1-1-1-1-1-1-1-1-1-1    linear of order 2
ρ52222002-1-1-1-2-200222-1-1-1-2-2-2-2111111    orthogonal lifted from D6
ρ6222200-1-12-1-2-200-1-1-12-1-11111-21-2111    orthogonal lifted from D6
ρ7222200-12-1-12200-1-1-1-12-1-1-1-1-1-12-1-12-1    orthogonal lifted from S3
ρ8222200-1-1-122200-1-1-1-1-12-1-1-1-1-1-1-12-12    orthogonal lifted from S3
ρ92-22-20022220000-2-22-2-2-20000000000    orthogonal lifted from D4
ρ10222200-1-12-12200-1-1-12-1-1-1-1-1-12-12-1-1-1    orthogonal lifted from S3
ρ11222200-12-1-1-2-200-1-1-1-12-111111-211-21    orthogonal lifted from D6
ρ122222002-1-1-12200222-1-1-12222-1-1-1-1-1-1    orthogonal lifted from S3
ρ13222200-1-1-12-2-200-1-1-1-1-121111111-21-2    orthogonal lifted from D6
ρ142-22-200-1-12-1000011-1-2113-33-30-3033-3    orthogonal lifted from D12
ρ152-22-2002-1-1-10000-2-2211100003-3-3-333    orthogonal lifted from D12
ρ162-22-200-1-1-12000011-111-2-33-33-3-33030    orthogonal lifted from D12
ρ172-22-200-1-12-1000011-1-211-33-33030-3-33    orthogonal lifted from D12
ρ182-22-200-12-1-1000011-11-21-33-3330-330-3    orthogonal lifted from D12
ρ192-22-2002-1-1-10000-2-221110000-3333-3-3    orthogonal lifted from D12
ρ202-22-200-12-1-1000011-11-213-33-3-303-303    orthogonal lifted from D12
ρ212-22-200-1-1-12000011-111-23-33-333-30-30    orthogonal lifted from D12
ρ2233-1-1-1-130003-1113-1-100033-1-1000000    orthogonal lifted from S4
ρ2333-1-11130003-1-1-13-1-100033-1-1000000    orthogonal lifted from S4
ρ2433-1-1-113000-311-13-1-1000-3-311000000    orthogonal lifted from C2×S4
ρ2533-1-11-13000-31-113-1-1000-3-311000000    orthogonal lifted from C2×S4
ρ2666-2-200-30006-200-311000-3-311000000    orthogonal lifted from C3⋊S4
ρ276-6-220060000000-62-20000000000000    orthogonal lifted from C4⋊S4
ρ2866-2-200-3000-6200-31100033-1-1000000    orthogonal lifted from C2×C3⋊S4
ρ296-6-2200-300000003-11000-33333-3000000    orthogonal faithful
ρ306-6-2200-300000003-1100033-33-33000000    orthogonal faithful

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

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

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

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

Matrix representation of C12⋊S4 in GL5(𝔽13)

87000
129000
00100
00010
00001
,
10000
01000
001200
001201
001210
,
10000
01000
000112
001012
000012
,
42000
98000
00010
00001
00100
,
112000
012000
00010
00100
00001

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

C12⋊S4 in GAP, Magma, Sage, TeX

C_{12}\rtimes S_4
% in TeX

G:=Group("C12:S4");
// GroupNames label

G:=SmallGroup(288,909);
// by ID

G=gap.SmallGroup(288,909);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-3,-2,2,85,36,451,1684,6053,782,3534,1350]);
// 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,e*a*e=a^-1,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

Export

Character table of C12⋊S4 in TeX

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