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G = C3×C3⋊C8order 72 = 23·32

Direct product of C3 and C3⋊C8

direct product, metacyclic, supersoluble, monomial, A-group

Aliases: C3×C3⋊C8, C3⋊C24, C6.C12, C323C8, C12.2C6, C12.8S3, C6.4Dic3, C4.2(C3×S3), (C3×C6).2C4, C2.(C3×Dic3), (C3×C12).3C2, SmallGroup(72,12)

Series: Derived Chief Lower central Upper central

C1C3 — C3×C3⋊C8
C1C3C6C12C3×C12 — C3×C3⋊C8
C3 — C3×C3⋊C8
C1C12

Generators and relations for C3×C3⋊C8
 G = < a,b,c | a3=b3=c8=1, ab=ba, ac=ca, cbc-1=b-1 >

2C3
2C6
3C8
2C12
3C24

Permutation representations of C3×C3⋊C8
On 24 points - transitive group 24T69
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 22 10)(3 11 23)(4 24 12)(5 13 17)(6 18 14)(7 15 19)(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), (1,9,21)(2,22,10)(3,11,23)(4,24,12)(5,13,17)(6,18,14)(7,15,19)(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), (1,9,21)(2,22,10)(3,11,23)(4,24,12)(5,13,17)(6,18,14)(7,15,19)(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)], [(1,9,21),(2,22,10),(3,11,23),(4,24,12),(5,13,17),(6,18,14),(7,15,19),(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(24,69);

C3×C3⋊C8 is a maximal subgroup of
C12.29D6  D6.Dic3  C12.31D6  C3⋊D24  D12.S3  C325SD16  C323Q16  S3×C24  He33C8  C9⋊C24  He34C8  SL2(𝔽3).Dic3
C3×C3⋊C8 is a maximal quotient of
He33C8  C9⋊C24

36 conjugacy classes

class 1  2 3A3B3C3D3E4A4B6A6B6C6D6E8A8B8C8D12A12B12C12D12E···12J24A···24H
order1233333446666688881212121212···1224···24
size11112221111222333311112···23···3

36 irreducible representations

dim11111111222222
type+++-
imageC1C2C3C4C6C8C12C24S3Dic3C3×S3C3⋊C8C3×Dic3C3×C3⋊C8
kernelC3×C3⋊C8C3×C12C3⋊C8C3×C6C12C32C6C3C12C6C4C3C2C1
# reps11222448112224

Matrix representation of C3×C3⋊C8 in GL2(𝔽13) generated by

90
09
,
90
03
,
05
10
G:=sub<GL(2,GF(13))| [9,0,0,9],[9,0,0,3],[0,1,5,0] >;

C3×C3⋊C8 in GAP, Magma, Sage, TeX

C_3\times C_3\rtimes C_8
% in TeX

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

G:=SmallGroup(72,12);
// by ID

G=gap.SmallGroup(72,12);
# by ID

G:=PCGroup([5,-2,-3,-2,-2,-3,30,42,1204]);
// Polycyclic

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

Export

Subgroup lattice of C3×C3⋊C8 in TeX

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