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## G = C3×C33⋊C4order 324 = 22·34

### Direct product of C3 and C33⋊C4

Aliases: C3×C33⋊C4, C3C4, AΣL1(𝔽81), C342C4, C336C12, C337Dic3, C322(C32⋊C4), C324(C3×Dic3), C3⋊(C3×C32⋊C4), C3⋊S3.(C3×S3), (C3×C3⋊S3).2S3, (C3×C3⋊S3).5C6, (C32×C3⋊S3).2C2, SmallGroup(324,162)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C33 — C3×C33⋊C4
 Chief series C1 — C3 — C33 — C3×C3⋊S3 — C32×C3⋊S3 — C3×C33⋊C4
 Lower central C33 — C3×C33⋊C4
 Upper central C1 — C3

Generators and relations for C3×C33⋊C4
G = < a,b,c,d,e | a3=b3=c3=d3=e4=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe-1=bc-1, cd=dc, ece-1=b-1c-1, ede-1=d-1 >

Subgroups: 424 in 96 conjugacy classes, 14 normal (all characteristic)
C1, C2, C3 [×2], C3 [×11], C4, S3 [×2], C6 [×3], C32 [×2], C32 [×36], Dic3, C12, C3×S3 [×8], C3⋊S3, C3×C6, C33 [×2], C33 [×11], C3×Dic3, C32⋊C4, S3×C32 [×2], C3×C3⋊S3 [×2], C3×C3⋊S3, C34, C3×C32⋊C4, C33⋊C4, C32×C3⋊S3, C3×C33⋊C4
Quotients: C1, C2, C3, C4, S3, C6, Dic3, C12, C3×S3, C3×Dic3, C32⋊C4, C3×C32⋊C4, C33⋊C4, C3×C33⋊C4

Permutation representations of C3×C33⋊C4
On 12 points - transitive group 12T131
Generators in S12
(1 9 8)(2 10 5)(3 11 6)(4 12 7)
(1 9 8)(2 5 10)(3 6 11)(4 12 7)
(1 8 9)(3 11 6)
(1 9 8)(2 5 10)(3 11 6)(4 7 12)
(1 2 3 4)(5 6 7 8)(9 10 11 12)

G:=sub<Sym(12)| (1,9,8)(2,10,5)(3,11,6)(4,12,7), (1,9,8)(2,5,10)(3,6,11)(4,12,7), (1,8,9)(3,11,6), (1,9,8)(2,5,10)(3,11,6)(4,7,12), (1,2,3,4)(5,6,7,8)(9,10,11,12)>;

G:=Group( (1,9,8)(2,10,5)(3,11,6)(4,12,7), (1,9,8)(2,5,10)(3,6,11)(4,12,7), (1,8,9)(3,11,6), (1,9,8)(2,5,10)(3,11,6)(4,7,12), (1,2,3,4)(5,6,7,8)(9,10,11,12) );

G=PermutationGroup([(1,9,8),(2,10,5),(3,11,6),(4,12,7)], [(1,9,8),(2,5,10),(3,6,11),(4,12,7)], [(1,8,9),(3,11,6)], [(1,9,8),(2,5,10),(3,11,6),(4,7,12)], [(1,2,3,4),(5,6,7,8),(9,10,11,12)])

G:=TransitiveGroup(12,131);

On 18 points - transitive group 18T123
Generators in S18
(1 3 5)(2 4 6)(7 17 14)(8 18 11)(9 15 12)(10 16 13)
(2 15 17)(4 12 14)(6 9 7)
(1 16 18)(2 15 17)(3 13 11)(4 12 14)(5 10 8)(6 9 7)
(1 3 5)(2 6 4)(7 14 17)(8 18 11)(9 12 15)(10 16 13)
(1 2)(3 4)(5 6)(7 8 9 10)(11 12 13 14)(15 16 17 18)

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

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

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

G:=TransitiveGroup(18,123);

Polynomial with Galois group C3×C33⋊C4 over ℚ
actionf(x)Disc(f)
12T131x12-69x10-72x9+1566x8+2466x7-14530x6-27216x5+49953x4+102474x3-21636x2-43272x+9616220·334·59·292·592·4992·6012·32394292

36 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E 3F ··· 3W 4A 4B 6A 6B 6C 6D 6E 12A 12B 12C 12D order 1 2 3 3 3 3 3 3 ··· 3 4 4 6 6 6 6 6 12 12 12 12 size 1 9 1 1 2 2 2 4 ··· 4 27 27 9 9 18 18 18 27 27 27 27

36 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 4 4 4 4 type + + + - + image C1 C2 C3 C4 C6 C12 S3 Dic3 C3×S3 C3×Dic3 C32⋊C4 C3×C32⋊C4 C33⋊C4 C3×C33⋊C4 kernel C3×C33⋊C4 C32×C3⋊S3 C33⋊C4 C34 C3×C3⋊S3 C33 C3×C3⋊S3 C33 C3⋊S3 C32 C32 C3 C3 C1 # reps 1 1 2 2 2 4 1 1 2 2 2 4 4 8

Matrix representation of C3×C33⋊C4 in GL4(𝔽7) generated by

 2 0 0 0 0 2 0 0 0 0 2 0 0 0 0 2
,
 3 2 4 3 4 5 5 6 3 3 6 1 0 0 0 1
,
 0 5 2 6 0 2 0 2 3 3 6 1 0 0 0 4
,
 3 6 3 2 6 3 4 2 0 0 2 0 0 0 0 4
,
 5 3 5 3 1 1 0 4 2 5 6 3 6 6 1 2
G:=sub<GL(4,GF(7))| [2,0,0,0,0,2,0,0,0,0,2,0,0,0,0,2],[3,4,3,0,2,5,3,0,4,5,6,0,3,6,1,1],[0,0,3,0,5,2,3,0,2,0,6,0,6,2,1,4],[3,6,0,0,6,3,0,0,3,4,2,0,2,2,0,4],[5,1,2,6,3,1,5,6,5,0,6,1,3,4,3,2] >;

C3×C33⋊C4 in GAP, Magma, Sage, TeX

C_3\times C_3^3\rtimes C_4
% in TeX

G:=Group("C3xC3^3:C4");
// GroupNames label

G:=SmallGroup(324,162);
// by ID

G=gap.SmallGroup(324,162);
# by ID

G:=PCGroup([6,-2,-3,-2,-3,3,-3,36,1443,111,1444,376,7781]);
// Polycyclic

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

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