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G = C4×C3⋊S4order 288 = 25·32

Direct product of C4 and C3⋊S4

direct product, non-abelian, soluble, monomial

Aliases: C4×C3⋊S4, C122S4, C32(C4×S4), (C4×A4)⋊2S3, A42(C4×S3), (C12×A4)⋊5C2, C6.29(C2×S4), (C2×A4).9D6, C6.7S45C2, (C22×C12)⋊3S3, (C22×C6).20D6, (C6×A4).14C22, C22⋊(C4×C3⋊S3), (C2×C6)⋊4(C4×S3), C2.1(C2×C3⋊S4), (C3×A4)⋊5(C2×C4), (C2×C3⋊S4).2C2, C23.2(C2×C3⋊S3), (C22×C4)⋊1(C3⋊S3), SmallGroup(288,908)

Series: Derived Chief Lower central Upper central

C1C22C3×A4 — C4×C3⋊S4
C1C22C2×C6C3×A4C6×A4C2×C3⋊S4 — C4×C3⋊S4
C3×A4 — C4×C3⋊S4
C1C4

Generators and relations for C4×C3⋊S4
 G = < a,b,c,d,e,f | a4=b3=c2=d2=e3=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, fbf=b-1, ece-1=fcf=cd=dc, ede-1=c, df=fd, fef=e-1 >

Subgroups: 784 in 144 conjugacy classes, 29 normal (18 characteristic)
C1, C2, C2 [×4], C3, C3 [×3], C4, C4 [×5], C22, C22 [×6], S3 [×8], C6, C6 [×5], C2×C4 [×7], D4 [×4], C23, C23, C32, Dic3 [×7], C12, C12 [×4], A4 [×3], D6 [×7], C2×C6, C2×C6 [×2], C42, C22⋊C4 [×2], C4⋊C4, C22×C4, C22×C4, C2×D4, C3⋊S3 [×2], C3×C6, C4×S3 [×5], C2×Dic3 [×3], C3⋊D4 [×4], C2×C12 [×2], S4 [×6], C2×A4 [×3], C22×S3, C22×C6, C4×D4, C3⋊Dic3, C3×C12, C3×A4, C2×C3⋊S3, C4×Dic3, Dic3⋊C4, D6⋊C4, C6.D4, A4⋊C4 [×3], C4×A4 [×3], S3×C2×C4, C2×C3⋊D4, C22×C12, C2×S4 [×3], C4×C3⋊S3, C3⋊S4 [×2], C6×A4, C4×C3⋊D4, C4×S4 [×3], C6.7S4, C12×A4, C2×C3⋊S4, C4×C3⋊S4
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×4], C2×C4, D6 [×4], C3⋊S3, C4×S3 [×4], S4, C2×C3⋊S3, C2×S4, C4×C3⋊S3, C3⋊S4, C4×S4, C2×C3⋊S4, C4×C3⋊S4

Smallest permutation representation of C4×C3⋊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 13 5)(2 14 6)(3 15 7)(4 16 8)(9 36 20)(10 33 17)(11 34 18)(12 35 19)(21 25 29)(22 26 30)(23 27 31)(24 28 32)
(1 3)(2 4)(5 7)(6 8)(13 15)(14 16)(21 23)(22 24)(25 27)(26 28)(29 31)(30 32)
(9 11)(10 12)(17 19)(18 20)(21 23)(22 24)(25 27)(26 28)(29 31)(30 32)(33 35)(34 36)
(1 23 35)(2 24 36)(3 21 33)(4 22 34)(5 31 12)(6 32 9)(7 29 10)(8 30 11)(13 27 19)(14 28 20)(15 25 17)(16 26 18)
(1 3)(2 4)(5 15)(6 16)(7 13)(8 14)(9 26)(10 27)(11 28)(12 25)(17 31)(18 32)(19 29)(20 30)(21 35)(22 36)(23 33)(24 34)

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

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

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

36 conjugacy classes

class 1 2A2B2C2D2E3A3B3C3D4A4B4C4D4E···4J6A6B6C6D6E6F12A12B12C12D12E···12J
order122222333344444···46666661212121212···12
size113318182888113318···1826688822668···8

36 irreducible representations

dim11111222222333666
type++++++++++++
imageC1C2C2C2C4S3S3D6D6C4×S3C4×S3S4C2×S4C4×S4C3⋊S4C2×C3⋊S4C4×C3⋊S4
kernelC4×C3⋊S4C6.7S4C12×A4C2×C3⋊S4C3⋊S4C4×A4C22×C12C2×A4C22×C6A4C2×C6C12C6C3C4C2C1
# reps11114313162224112

Matrix representation of C4×C3⋊S4 in GL5(𝔽13)

80000
08000
001200
000120
000012
,
112000
51000
00100
00010
00001
,
10000
01000
00100
0012120
001012
,
10000
01000
001200
000120
001201
,
10000
01000
00120
0001212
00010
,
211000
811000
0012110
00010
0001212

G:=sub<GL(5,GF(13))| [8,0,0,0,0,0,8,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[11,5,0,0,0,2,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,12,1,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,1,0,0,0,0,0,12,0,12,0,0,0,12,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,2,12,1,0,0,0,12,0],[2,8,0,0,0,11,11,0,0,0,0,0,12,0,0,0,0,11,1,12,0,0,0,0,12] >;

C4×C3⋊S4 in GAP, Magma, Sage, TeX

C_4\times C_3\rtimes S_4
% in TeX

G:=Group("C4xC3:S4");
// GroupNames label

G:=SmallGroup(288,908);
// by ID

G=gap.SmallGroup(288,908);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-3,-2,2,36,451,1684,6053,782,3534,1350]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^4=b^3=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,a*f=f*a,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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