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G = C2×S3×C3.A4  order 432 = 24·33

Direct product of C2, S3 and C3.A4

direct product, metabelian, soluble, monomial, A-group

Aliases: C2×S3×C3.A4, (S3×C23)⋊C9, (C22×C6)⋊C18, (C22×S3)⋊C18, (S3×C6).2A4, C6.19(S3×A4), C23⋊2(S3×C9), C22⋊2(S3×C18), (C2×C62).8C6, C62.14(C2×C6), C32.2(C22×A4), C6⋊(C2×C3.A4), (S3×C2×C6).C6, (C2×C6)⋊(C2×C18), C3.4(C2×S3×A4), (C3×S3).(C2×A4), C3⋊(C22×C3.A4), (S3×C22×C6).C3, (C6×C3.A4)⋊1C2, (C2×C6).17(S3×C6), (C3×C6).14(C2×A4), (C3×C3.A4)⋊2C22, (C22×C6).19(C3×S3), SmallGroup(432,541)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C6 — C2×S3×C3.A4
Chief series
C1 — C3 — C2×C6 — C62 — C3×C3.A4 — S3×C3.A4 — C2×S3×C3.A4
Lower central
C2×C6 — C2×S3×C3.A4
Upper central
C1 — C6

Generators and relations for C2×S3×C3.A4
 G = < a,b,c,d,e,f,g | a2=b3=c2=d3=e2=f2=1, g3=d, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, cbc=b-1, bd=db, be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, de=ed, df=fd, dg=gd, geg-1=ef=fe, gfg-1=e >

Subgroups: 604 in 148 conjugacy classes, 35 normal (25 characteristic)
C1, C2, C2, C3, C3, C22, C22, S3, S3, C6, C6, C23, C23, C9, C32, D6, D6, C2×C6, C2×C6, C24, C18, C3×S3, C3×S3, C3×C6, C3×C6, C22×S3, C22×S3, C22×C6, C22×C6, C3×C9, C3.A4, C3.A4, C2×C18, S3×C6, S3×C6, C62, C62, S3×C23, C23×C6, S3×C9, C3×C18, C2×C3.A4, C2×C3.A4, S3×C2×C6, S3×C2×C6, C2×C62, C3×C3.A4, S3×C18, C22×C3.A4, S3×C22×C6, S3×C3.A4, C6×C3.A4, C2×S3×C3.A4
Quotients: C1, C2, C3, C22, S3, C6, C9, A4, D6, C2×C6, C18, C3×S3, C2×A4, C3.A4, C2×C18, S3×C6, C22×A4, S3×C9, C2×C3.A4, S3×A4, S3×C18, C22×C3.A4, C2×S3×A4, S3×C3.A4, C2×S3×C3.A4

Smallest permutation representation of C2×S3×C3.A4
►On 36 points
Generators in S36
(1 20)(2 21)(3 22)(4 23)(5 24)(6 25)(7 26)(8 27)(9 19)(10 31)(11 32)(12 33)(13 34)(14 35)(15 36)(16 28)(17 29)(18 30)

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

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

(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 20)(3 22)(4 23)(6 25)(7 26)(9 19)(10 31)(12 33)(13 34)(15 36)(16 28)(18 30)

(1 20)(2 21)(4 23)(5 24)(7 26)(8 27)(10 31)(11 32)(13 34)(14 35)(16 28)(17 29)

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

72 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B3C3D3E6A6B6C6D6E6F···6M6N···6S6T6U6V6W9A···9F9G···9L18A···18F18G···18L18M···18X
order1222222233333666666···66···666669···99···918···1818···1818···18
size1133339911222112223···36···699994···48···84···48···812···12

72 irreducible representations

dim1111111112222223333336666
type++++++++++
imageC1C2C2C3C6C6C9C18C18S3D6C3×S3S3×C6S3×C9S3×C18A4C2×A4C2×A4C3.A4C2×C3.A4C2×C3.A4S3×A4C2×S3×A4S3×C3.A4C2×S3×C3.A4
kernelC2×S3×C3.A4S3×C3.A4C6×C3.A4S3×C22×C6S3×C2×C6C2×C62S3×C23C22×S3C22×C6C2×C3.A4C3.A4C22×C6C2×C6C23C22S3×C6C3×S3C3×C6D6S3C6C6C3C2C1
# reps12124261261122661212421122

Matrix representation of C2×S3×C3.A4 ►in GL5(𝔽19)

180000
018000
00100
00010
00001
,
110000
07000
00100
00010
00001
,
018000
180000
00100
00010
00001
,
10000
01000
00700
00070
00007
,
10000
01000
001800
000180
00001
,
10000
01000
001800
00010
000018
,
10000
01000
00060
00006
00600

G:=sub<GL(5,GF(19))| [18,0,0,0,0,0,18,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[11,0,0,0,0,0,7,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[0,18,0,0,0,18,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,7,0,0,0,0,0,7,0,0,0,0,0,7],[1,0,0,0,0,0,1,0,0,0,0,0,18,0,0,0,0,0,18,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,18,0,0,0,0,0,1,0,0,0,0,0,18],[1,0,0,0,0,0,1,0,0,0,0,0,0,0,6,0,0,6,0,0,0,0,0,6,0] >;
 

C2×S3×C3.A4 in GAP, Magma, Sage, TeX 

C_2\times S_3\times C_3.A_4
 
% in TeX
 
G:=Group("C2xS3xC3.A4");
 
// GroupNames label
 
G:=SmallGroup(432,541);
 
// by ID
 
G=gap.SmallGroup(432,541);
 
# by ID
 
G:=PCGroup([7,-2,-2,-3,-3,-2,2,-3,79,963,397,14118]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^3=c^2=d^3=e^2=f^2=1,g^3=d,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,c*e=e*c,c*f=f*c,c*g=g*c,d*e=e*d,d*f=f*d,d*g=g*d,g*e*g^-1=e*f=f*e,g*f*g^-1=e>;
 
// generators/relations
 

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