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G = C3×C322C8order 216 = 23·33

Direct product of C3 and C322C8

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

Aliases: C3×C322C8, C333C8, C323C24, (C3×C6).2C12, C6.4(C32⋊C4), C3⋊Dic3.2C6, (C32×C6).1C4, C2.(C3×C32⋊C4), (C3×C3⋊Dic3).1C2, SmallGroup(216,117)

Series: Derived Chief Lower central Upper central

C1C32 — C3×C322C8
C1C32C3×C6C3⋊Dic3C3×C3⋊Dic3 — C3×C322C8
C32 — C3×C322C8
C1C6

Generators and relations for C3×C322C8
 G = < a,b,c,d | a3=b3=c3=d8=1, ab=ba, ac=ca, ad=da, dcd-1=bc=cb, dbd-1=b-1c >

2C3
2C3
4C3
4C3
9C4
2C6
2C6
4C6
4C6
2C32
2C32
4C32
4C32
9C8
6Dic3
6Dic3
9C12
2C3×C6
2C3×C6
4C3×C6
4C3×C6
9C24
6C3×Dic3
6C3×Dic3

Permutation representations of C3×C322C8
On 24 points - transitive group 24T554
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)
(2 10 22)(4 24 12)(6 14 18)(8 20 16)
(1 9 21)(2 10 22)(3 23 11)(4 24 12)(5 13 17)(6 14 18)(7 19 15)(8 20 16)
(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), (2,10,22)(4,24,12)(6,14,18)(8,20,16), (1,9,21)(2,10,22)(3,23,11)(4,24,12)(5,13,17)(6,14,18)(7,19,15)(8,20,16), (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), (2,10,22)(4,24,12)(6,14,18)(8,20,16), (1,9,21)(2,10,22)(3,23,11)(4,24,12)(5,13,17)(6,14,18)(7,19,15)(8,20,16), (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)], [(2,10,22),(4,24,12),(6,14,18),(8,20,16)], [(1,9,21),(2,10,22),(3,23,11),(4,24,12),(5,13,17),(6,14,18),(7,19,15),(8,20,16)], [(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(24,554);

C3×C322C8 is a maximal subgroup of   C6.F9  C335(C2×C8)  C33⋊M4(2)  C332M4(2)  C322D24  C338SD16  C333Q16

36 conjugacy classes

class 1  2 3A3B3C···3H4A4B6A6B6C···6H8A8B8C8D12A12B12C12D24A···24H
order12333···344666···688881212121224···24
size11114···499114···4999999999···9

36 irreducible representations

dim111111114444
type+++-
imageC1C2C3C4C6C8C12C24C32⋊C4C322C8C3×C32⋊C4C3×C322C8
kernelC3×C322C8C3×C3⋊Dic3C322C8C32×C6C3⋊Dic3C33C3×C6C32C6C3C2C1
# reps112224482244

Matrix representation of C3×C322C8 in GL4(𝔽7) generated by

4000
0400
0040
0004
,
5156
5032
4433
0564
,
6463
3615
4631
0430
,
4043
2410
3253
3261
G:=sub<GL(4,GF(7))| [4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[5,5,4,0,1,0,4,5,5,3,3,6,6,2,3,4],[6,3,4,0,4,6,6,4,6,1,3,3,3,5,1,0],[4,2,3,3,0,4,2,2,4,1,5,6,3,0,3,1] >;

C3×C322C8 in GAP, Magma, Sage, TeX

C_3\times C_3^2\rtimes_2C_8
% in TeX

G:=Group("C3xC3^2:2C8");
// GroupNames label

G:=SmallGroup(216,117);
// by ID

G=gap.SmallGroup(216,117);
# by ID

G:=PCGroup([6,-2,-3,-2,-2,-3,3,36,50,5044,256,6917,881]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^3=c^3=d^8=1,a*b=b*a,a*c=c*a,a*d=d*a,d*c*d^-1=b*c=c*b,d*b*d^-1=b^-1*c>;
// generators/relations

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Subgroup lattice of C3×C322C8 in TeX

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