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G = C3×C62.C4  order 432 = 24·33

Direct product of C3 and C62.C4

direct product, metabelian, soluble, monomial

Aliases: C3×C62.C4, C62.8C12, C33⋊11M4(2), C32⋊2C8⋊4C6, (C3×C62).2C4, C3⋊Dic3.8C12, C32⋊6(C3×M4(2)), C2.6(C6×C32⋊C4), C22.(C3×C32⋊C4), C6.24(C2×C32⋊C4), (C3×C6).15(C2×C12), (C2×C6).3(C32⋊C4), (C3×C32⋊2C8)⋊10C2, (C6×C3⋊Dic3).17C2, (C3×C3⋊Dic3).13C4, C3⋊Dic3.15(C2×C6), (C2×C3⋊Dic3).11C6, (C32×C6).13(C2×C4), (C3×C3⋊Dic3).40C22, SmallGroup(432,633)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3×C6 — C3×C62.C4
Chief series
C1 — C32 — C3×C6 — C3⋊Dic3 — C3×C3⋊Dic3 — C3×C32⋊2C8 — C3×C62.C4
Lower central
C32 — C3×C6 — C3×C62.C4
Upper central
C1 — C6 — C2×C6

Generators and relations for C3×C62.C4
 G = < a,b,c,d | a3=b6=c6=1, d4=c3, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1c, dcd-1=b4c >

Subgroups: 332 in 92 conjugacy classes, 24 normal (20 characteristic)
C1, C2, C2, C3, C3, C4, C22, C6, C6, C8, C2×C4, C32, C32, Dic3, C12, C2×C6, C2×C6, M4(2), C3×C6, C3×C6, C24, C2×Dic3, C2×C12, C33, C3×Dic3, C3⋊Dic3, C62, C62, C3×M4(2), C32×C6, C32×C6, C32⋊2C8, C6×Dic3, C2×C3⋊Dic3, C3×C3⋊Dic3, C3×C62, C62.C4, C3×C32⋊2C8, C6×C3⋊Dic3, C3×C62.C4
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C12, C2×C6, M4(2), C2×C12, C32⋊C4, C3×M4(2), C2×C32⋊C4, C3×C32⋊C4, C62.C4, C6×C32⋊C4, C3×C62.C4

Permutation representations of C3×C62.C4
►On 24 points - transitive group 24T1289
Generators in S24
(1 9 21)(2 10 22)(3 11 23)(4 12 24)(5 13 17)(6 14 18)(7 15 19)(8 16 20)

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

(1 17 9 5 21 13)(2 6)(3 15 23 7 11 19)(4 8)(10 14)(12 16)(18 22)(20 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)
 
G:=sub<Sym(24)| (1,9,21)(2,10,22)(3,11,23)(4,12,24)(5,13,17)(6,14,18)(7,15,19)(8,16,20), (1,9,21)(2,14,22,6,10,18)(3,23,11)(4,20,12,8,24,16)(5,13,17)(7,19,15), (1,17,9,5,21,13)(2,6)(3,15,23,7,11,19)(4,8)(10,14)(12,16)(18,22)(20,24), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)>;
 
G:=Group( (1,9,21)(2,10,22)(3,11,23)(4,12,24)(5,13,17)(6,14,18)(7,15,19)(8,16,20), (1,9,21)(2,14,22,6,10,18)(3,23,11)(4,20,12,8,24,16)(5,13,17)(7,19,15), (1,17,9,5,21,13)(2,6)(3,15,23,7,11,19)(4,8)(10,14)(12,16)(18,22)(20,24), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24) );
 
G=PermutationGroup([[(1,9,21),(2,10,22),(3,11,23),(4,12,24),(5,13,17),(6,14,18),(7,15,19),(8,16,20)], [(1,9,21),(2,14,22,6,10,18),(3,23,11),(4,20,12,8,24,16),(5,13,17),(7,19,15)], [(1,17,9,5,21,13),(2,6),(3,15,23,7,11,19),(4,8),(10,14),(12,16),(18,22),(20,24)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24)]])
 
G:=TransitiveGroup(24,1289);
 

54 conjugacy classes

class 1 2A2B3A3B3C···3H4A4B4C6A6B6C6D6E···6V8A8B8C8D12A12B12C12D12E12F24A···24H
order122333···344466666···6888812121212121224···24
size112114···4991811224···4181818189999181818···18

54 irreducible representations

dim111111111122444444
type+++++-
imageC1C2C2C3C4C4C6C6C12C12M4(2)C3×M4(2)C32⋊C4C2×C32⋊C4C3×C32⋊C4C62.C4C6×C32⋊C4C3×C62.C4
kernelC3×C62.C4C3×C32⋊2C8C6×C3⋊Dic3C62.C4C3×C3⋊Dic3C3×C62C32⋊2C8C2×C3⋊Dic3C3⋊Dic3C62C33C32C2×C6C6C22C3C2C1
# reps121222424424224448

Matrix representation of C3×C62.C4 ►in GL4(𝔽7) generated by

4000
0400
0040
0004
,
2604
4051
4406
0005
,
4534
3221
4416
0006
,
6163
6663
5214
3341
G:=sub<GL(4,GF(7))| [4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[2,4,4,0,6,0,4,0,0,5,0,0,4,1,6,5],[4,3,4,0,5,2,4,0,3,2,1,0,4,1,6,6],[6,6,5,3,1,6,2,3,6,6,1,4,3,3,4,1] >;
 

C3×C62.C4 in GAP, Magma, Sage, TeX 

C_3\times C_6^2.C_4
 
% in TeX
 
G:=Group("C3xC6^2.C4");
 
// GroupNames label
 
G:=SmallGroup(432,633);
 
// by ID
 
G=gap.SmallGroup(432,633);
 
# by ID
 
G:=PCGroup([7,-2,-2,-3,-2,-2,-3,3,84,365,80,14117,362,18822,1203]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^3=b^6=c^6=1,d^4=c^3,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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