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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, C12.10S4, C23.2D18, C3.(C4×S4), C22⋊(C4×D9), C6.17(C2×S4), (C22×C4)⋊1D9, C6.S42C2, (C22×C12).6S3, (C22×C6).14D6, (C2×C6).(C4×S3), (C2×C3.S4).C2, (C4×C3.A4)⋊2C2, C3.A41(C2×C4), C2.1(C2×C3.S4), (C2×C3.A4).2C22, SmallGroup(288,333)

Series: Derived Chief Lower central Upper central

C1C22C3.A4 — C4×C3.S4
C1C22C2×C6C3.A4C2×C3.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=f2=1, e3=b, 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=b-1e2 >

Subgroups: 514 in 96 conjugacy classes, 20 normal (18 characteristic)
C1, C2, C2 [×4], C3, C4, C4 [×5], C22, C22 [×6], S3 [×2], C6, C6 [×2], C2×C4 [×7], D4 [×4], C23, C23, C9, Dic3 [×4], C12, C12, D6 [×4], C2×C6, C2×C6 [×2], C42, C22⋊C4 [×2], C4⋊C4, C22×C4, C22×C4, C2×D4, D9 [×2], C18, C4×S3 [×2], C2×Dic3 [×3], C3⋊D4 [×4], C2×C12 [×2], C22×S3, C22×C6, C4×D4, Dic9, C36, C3.A4, D18, C4×Dic3, Dic3⋊C4, D6⋊C4, C6.D4, S3×C2×C4, C2×C3⋊D4, C22×C12, C4×D9, C3.S4 [×2], C2×C3.A4, C4×C3⋊D4, C6.S4, C4×C3.A4, C2×C3.S4, C4×C3.S4
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D6, D9, C4×S3, S4, D18, C2×S4, C4×D9, C3.S4, C4×S4, C2×C3.S4, C4×C3.S4

Smallest permutation representation of C4×C3.S4
On 36 points
Generators in S36
(1 11 22 34)(2 12 23 35)(3 13 24 36)(4 14 25 28)(5 15 26 29)(6 16 27 30)(7 17 19 31)(8 18 20 32)(9 10 21 33)
(1 4 7)(2 5 8)(3 6 9)(10 13 16)(11 14 17)(12 15 18)(19 22 25)(20 23 26)(21 24 27)(28 31 34)(29 32 35)(30 33 36)
(1 22)(2 23)(4 25)(5 26)(7 19)(8 20)(11 34)(12 35)(14 28)(15 29)(17 31)(18 32)
(2 23)(3 24)(5 26)(6 27)(8 20)(9 21)(10 33)(12 35)(13 36)(15 29)(16 30)(18 32)
(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 22)(2 21)(3 20)(4 19)(5 27)(6 26)(7 25)(8 24)(9 23)(10 35)(11 34)(12 33)(13 32)(14 31)(15 30)(16 29)(17 28)(18 36)

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

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

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

36 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D4E···4J6A6B6C9A9B9C12A12B12C12D18A18B18C36A···36F
order122222344444···46669991212121218181836···36
size113318182113318···1826688822668888···8

36 irreducible representations

dim11111222222333666
type++++++++++++
imageC1C2C2C2C4S3D6D9C4×S3D18C4×D9S4C2×S4C4×S4C3.S4C2×C3.S4C4×C3.S4
kernelC4×C3.S4C6.S4C4×C3.A4C2×C3.S4C3.S4C22×C12C22×C6C22×C4C2×C6C23C22C12C6C3C4C2C1
# reps11114113236224112

Matrix representation of C4×C3.S4 in GL5(𝔽37)

60000
06000
003600
000360
000036
,
3636000
10000
00100
00010
00001
,
10000
01000
003600
00110
000036
,
10000
01000
003600
000360
003601
,
266000
3120000
001035
00001
0003636
,
10000
3636000
0036350
00010
0003636

G:=sub<GL(5,GF(37))| [6,0,0,0,0,0,6,0,0,0,0,0,36,0,0,0,0,0,36,0,0,0,0,0,36],[36,1,0,0,0,36,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,36,1,0,0,0,0,1,0,0,0,0,0,36],[1,0,0,0,0,0,1,0,0,0,0,0,36,0,36,0,0,0,36,0,0,0,0,0,1],[26,31,0,0,0,6,20,0,0,0,0,0,1,0,0,0,0,0,0,36,0,0,35,1,36],[1,36,0,0,0,0,36,0,0,0,0,0,36,0,0,0,0,35,1,36,0,0,0,0,36] >;

C4×C3.S4 in GAP, Magma, Sage, TeX

C_4\times C_3.S_4
% in TeX

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

G:=SmallGroup(288,333);
// by ID

G=gap.SmallGroup(288,333);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-3,-2,2,36,1123,192,1684,6053,782,3534,1350]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^4=b^3=c^2=d^2=f^2=1,e^3=b,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=b^-1*e^2>;
// generators/relations

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