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## G = C3×C3⋊C8order 72 = 23·32

### Direct product of C3 and C3⋊C8

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

 Derived series C1 — C3 — C3×C3⋊C8
 Chief series C1 — C3 — C6 — C12 — C3×C12 — C3×C3⋊C8
 Lower central C3 — C3×C3⋊C8
 Upper central C1 — C12

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

Permutation representations of C3×C3⋊C8
On 24 points - transitive group 24T69
Generators in S24
(1 15 21)(2 16 22)(3 9 23)(4 10 24)(5 11 17)(6 12 18)(7 13 19)(8 14 20)
(1 15 21)(2 22 16)(3 9 23)(4 24 10)(5 11 17)(6 18 12)(7 13 19)(8 20 14)
(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)| (1,15,21)(2,16,22)(3,9,23)(4,10,24)(5,11,17)(6,12,18)(7,13,19)(8,14,20), (1,15,21)(2,22,16)(3,9,23)(4,24,10)(5,11,17)(6,18,12)(7,13,19)(8,20,14), (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( (1,15,21)(2,16,22)(3,9,23)(4,10,24)(5,11,17)(6,12,18)(7,13,19)(8,14,20), (1,15,21)(2,22,16)(3,9,23)(4,24,10)(5,11,17)(6,18,12)(7,13,19)(8,20,14), (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([[(1,15,21),(2,16,22),(3,9,23),(4,10,24),(5,11,17),(6,12,18),(7,13,19),(8,14,20)], [(1,15,21),(2,22,16),(3,9,23),(4,24,10),(5,11,17),(6,18,12),(7,13,19),(8,20,14)], [(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 3A 3B 3C 3D 3E 4A 4B 6A 6B 6C 6D 6E 8A 8B 8C 8D 12A 12B 12C 12D 12E ··· 12J 24A ··· 24H order 1 2 3 3 3 3 3 4 4 6 6 6 6 6 8 8 8 8 12 12 12 12 12 ··· 12 24 ··· 24 size 1 1 1 1 2 2 2 1 1 1 1 2 2 2 3 3 3 3 1 1 1 1 2 ··· 2 3 ··· 3

36 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + - image C1 C2 C3 C4 C6 C8 C12 C24 S3 Dic3 C3×S3 C3⋊C8 C3×Dic3 C3×C3⋊C8 kernel C3×C3⋊C8 C3×C12 C3⋊C8 C3×C6 C12 C32 C6 C3 C12 C6 C4 C3 C2 C1 # reps 1 1 2 2 2 4 4 8 1 1 2 2 2 4

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

 9 0 0 9
,
 9 0 0 3
,
 0 5 1 0
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

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