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G = C3xC62:C4order 432 = 24·33

Direct product of C3 and C62:C4

direct product, metabelian, soluble, monomial

Aliases: C3xC62:C4, C62:8C12, (C3xC62):1C4, C33:6(C22:C4), (C6xC3:S3):5C4, (C2xC3:S3):5C12, (C6xC32:C4):4C2, (C2xC32:C4):2C6, C3:S3.5(C3xD4), C2.7(C6xC32:C4), (C2xC6):1(C32:C4), (C3xC3:S3).16D4, C6.25(C2xC32:C4), C22:2(C3xC32:C4), (C3xC6).16(C2xC12), (C22xC3:S3).6C6, C32:4(C3xC22:C4), (C6xC3:S3).40C22, (C32xC6).14(C2xC4), (C2xC6xC3:S3).5C2, (C2xC3:S3).15(C2xC6), SmallGroup(432,634)

Series: Derived Chief Lower central Upper central

C1C3xC6 — C3xC62:C4
C1C32C3xC6C2xC3:S3C6xC3:S3C6xC32:C4 — C3xC62:C4
C32C3xC6 — C3xC62:C4
C1C6C2xC6

Generators and relations for C3xC62:C4
 G = < a,b,c,d | a3=b6=c6=d4=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1c, dcd-1=b4c >

Subgroups: 764 in 152 conjugacy classes, 28 normal (20 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C2xC4, C23, C32, C32, C12, D6, C2xC6, C2xC6, C22:C4, C3xS3, C3:S3, C3:S3, C3xC6, C3xC6, C2xC12, C22xS3, C22xC6, C33, C32:C4, S3xC6, C2xC3:S3, C2xC3:S3, C62, C62, C3xC22:C4, C3xC3:S3, C3xC3:S3, C32xC6, C32xC6, C2xC32:C4, S3xC2xC6, C22xC3:S3, C3xC32:C4, C6xC3:S3, C6xC3:S3, C3xC62, C62:C4, C6xC32:C4, C2xC6xC3:S3, C3xC62:C4
Quotients: C1, C2, C3, C4, C22, C6, C2xC4, D4, C12, C2xC6, C22:C4, C2xC12, C3xD4, C32:C4, C3xC22:C4, C2xC32:C4, C3xC32:C4, C62:C4, C6xC32:C4, C3xC62:C4

Permutation representations of C3xC62:C4
On 24 points - transitive group 24T1288
Generators in S24
(1 6 7)(2 5 8)(3 10 11)(4 9 12)(13 15 14)(16 17 18)(19 21 20)(22 23 24)
(1 2)(3 4)(5 6)(7 8)(9 10)(11 12)(13 14 15)(16 17 18)(19 20 21)(22 23 24)
(1 8 6 2 7 5)(3 9 11 4 10 12)(13 21 15 20 14 19)(16 23 18 22 17 24)
(1 21 12 16)(2 14 11 22)(3 23 5 13)(4 17 6 20)(7 19 9 18)(8 15 10 24)

G:=sub<Sym(24)| (1,6,7)(2,5,8)(3,10,11)(4,9,12)(13,15,14)(16,17,18)(19,21,20)(22,23,24), (1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24), (1,8,6,2,7,5)(3,9,11,4,10,12)(13,21,15,20,14,19)(16,23,18,22,17,24), (1,21,12,16)(2,14,11,22)(3,23,5,13)(4,17,6,20)(7,19,9,18)(8,15,10,24)>;

G:=Group( (1,6,7)(2,5,8)(3,10,11)(4,9,12)(13,15,14)(16,17,18)(19,21,20)(22,23,24), (1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24), (1,8,6,2,7,5)(3,9,11,4,10,12)(13,21,15,20,14,19)(16,23,18,22,17,24), (1,21,12,16)(2,14,11,22)(3,23,5,13)(4,17,6,20)(7,19,9,18)(8,15,10,24) );

G=PermutationGroup([[(1,6,7),(2,5,8),(3,10,11),(4,9,12),(13,15,14),(16,17,18),(19,21,20),(22,23,24)], [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,14,15),(16,17,18),(19,20,21),(22,23,24)], [(1,8,6,2,7,5),(3,9,11,4,10,12),(13,21,15,20,14,19),(16,23,18,22,17,24)], [(1,21,12,16),(2,14,11,22),(3,23,5,13),(4,17,6,20),(7,19,9,18),(8,15,10,24)]])

G:=TransitiveGroup(24,1288);

54 conjugacy classes

class 1 2A2B2C2D2E3A3B3C···3H4A4B4C4D6A6B6C6D6E···6V6W6X6Y6Z6AA6AB12A···12H
order122222333···3444466666···666666612···12
size1129918114···41818181811224···49999181818···18

54 irreducible representations

dim111111111122444444
type+++++++
imageC1C2C2C3C4C4C6C6C12C12D4C3xD4C32:C4C2xC32:C4C3xC32:C4C62:C4C6xC32:C4C3xC62:C4
kernelC3xC62:C4C6xC32:C4C2xC6xC3:S3C62:C4C6xC3:S3C3xC62C2xC32:C4C22xC3:S3C2xC3:S3C62C3xC3:S3C3:S3C2xC6C6C22C3C2C1
# reps121222424424224448

Matrix representation of C3xC62:C4 in GL4(F7) generated by

4000
0400
0040
0004
,
5461
0456
3034
6662
,
1301
5646
1134
1503
,
5141
6121
5030
1635
G:=sub<GL(4,GF(7))| [4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[5,0,3,6,4,4,0,6,6,5,3,6,1,6,4,2],[1,5,1,1,3,6,1,5,0,4,3,0,1,6,4,3],[5,6,5,1,1,1,0,6,4,2,3,3,1,1,0,5] >;

C3xC62:C4 in GAP, Magma, Sage, TeX

C_3\times C_6^2\rtimes C_4
% in TeX

G:=Group("C3xC6^2:C4");
// GroupNames label

G:=SmallGroup(432,634);
// by ID

G=gap.SmallGroup(432,634);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,3,84,365,14117,362,18822,1203]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^6=c^6=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1*c,d*c*d^-1=b^4*c>;
// generators/relations

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