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G = C3⋊S3.2D12order 432 = 24·33

1st non-split extension by C3⋊S3 of D12 acting via D12/C6=C22

non-abelian, soluble, monomial

Aliases: C6.12S3≀C2, C3⋊S3.2D12, C6.D64S3, (C32×C6).6D4, C323(D6⋊C4), C324D61C4, C333(C22⋊C4), C2.1(C33⋊D4), C31(S32⋊C4), C3⋊S3.2(C4×S3), (C3×C3⋊S3).9D4, (C2×C3⋊S3).10D6, (C2×C33⋊C4)⋊2C2, (C3×C6.D6)⋊8C2, (C6×C3⋊S3).6C22, (C3×C6).12(C3⋊D4), (C2×C324D6).1C2, (C3×C3⋊S3).8(C2×C4), SmallGroup(432,579)

Series: Derived Chief Lower central Upper central

C1C32C3×C3⋊S3 — C3⋊S3.2D12
C1C3C33C3×C3⋊S3C6×C3⋊S3C2×C324D6 — C3⋊S3.2D12
C33C3×C3⋊S3 — C3⋊S3.2D12
C1C2

Generators and relations for C3⋊S3.2D12
 G = < a,b,c,d,e | a3=b3=c2=d12=1, e2=c, ab=ba, cac=ebe-1=a-1, ad=da, eae-1=b, cbc=dbd-1=b-1, cd=dc, ce=ec, ede-1=cd-1 >

Subgroups: 1012 in 132 conjugacy classes, 21 normal (19 characteristic)
C1, C2, C2 [×4], C3, C3 [×4], C4 [×2], C22 [×5], S3 [×10], C6, C6 [×8], C2×C4 [×2], C23, C32, C32 [×4], Dic3 [×2], C12 [×3], D6 [×12], C2×C6 [×2], C22⋊C4, C3×S3 [×10], C3⋊S3 [×2], C3⋊S3 [×2], C3×C6, C3×C6 [×4], C4×S3, C2×Dic3, C2×C12, C22×S3 [×2], C33, C3×Dic3 [×3], C3×C12, C32⋊C4, S32 [×7], S3×C6 [×5], C2×C3⋊S3, C2×C3⋊S3, D6⋊C4, C3×C3⋊S3 [×2], C3×C3⋊S3 [×2], C32×C6, C6.D6, S3×C12, C2×C32⋊C4, C2×S32 [×2], C32×Dic3, C33⋊C4, C324D6 [×2], C324D6, C6×C3⋊S3, C6×C3⋊S3, S32⋊C4, C3×C6.D6, C2×C33⋊C4, C2×C324D6, C3⋊S3.2D12
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], D6, C22⋊C4, C4×S3, D12, C3⋊D4, D6⋊C4, S3≀C2, S32⋊C4, C33⋊D4, C3⋊S3.2D12

Permutation representations of C3⋊S3.2D12
On 24 points - transitive group 24T1313
Generators in S24
(1 5 9)(2 6 10)(3 7 11)(4 8 12)(13 21 17)(14 22 18)(15 23 19)(16 24 20)
(1 5 9)(2 10 6)(3 7 11)(4 12 8)(13 21 17)(14 18 22)(15 23 19)(16 20 24)
(1 17)(2 18)(3 19)(4 20)(5 21)(6 22)(7 23)(8 24)(9 13)(10 14)(11 15)(12 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)
(1 22 17 6)(2 5 18 21)(3 20 19 4)(7 16 23 12)(8 11 24 15)(9 14 13 10)

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

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

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

G:=TransitiveGroup(24,1313);

36 conjugacy classes

class 1 2A2B2C2D2E3A3B3C3D3E3F4A4B4C4D6A6B6C6D6E6F6G6H6I6J12A12B12C12D12E···12J
order122222333333444466666666661212121212···12
size1199181824444866545424444818183636666612···12

36 irreducible representations

dim1111122222224444488
type+++++++++++++
imageC1C2C2C2C4S3D4D4D6C4×S3D12C3⋊D4S3≀C2S32⋊C4S32⋊C4C33⋊D4C3⋊S3.2D12C33⋊D4C3⋊S3.2D12
kernelC3⋊S3.2D12C3×C6.D6C2×C33⋊C4C2×C324D6C324D6C6.D6C3×C3⋊S3C32×C6C2×C3⋊S3C3⋊S3C3⋊S3C3×C6C6C3C3C2C1C2C1
# reps1111411112224224411

Matrix representation of C3⋊S3.2D12 in GL6(𝔽13)

100000
010000
00120120
000001
001000
00012012
,
100000
010000
000010
000001
00120120
00012012
,
100000
010000
001000
000100
00120120
00012012
,
10100000
370000
0001200
0012000
000101
001010
,
1060000
330000
000100
001000
00012012
000010

G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,1,0,0,0,0,0,0,12,0,0,12,0,0,0,0,0,0,1,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,1,0,12,0,0,0,0,1,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,12,0,0,0,0,1,0,12,0,0,0,0,12,0,0,0,0,0,0,12],[10,3,0,0,0,0,10,7,0,0,0,0,0,0,0,12,0,1,0,0,12,0,1,0,0,0,0,0,0,1,0,0,0,0,1,0],[10,3,0,0,0,0,6,3,0,0,0,0,0,0,0,1,0,0,0,0,1,0,12,0,0,0,0,0,0,1,0,0,0,0,12,0] >;

C3⋊S3.2D12 in GAP, Magma, Sage, TeX

C_3\rtimes S_3._2D_{12}
% in TeX

G:=Group("C3:S3.2D12");
// GroupNames label

G:=SmallGroup(432,579);
// by ID

G=gap.SmallGroup(432,579);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,85,64,1684,571,298,677,1027,14118]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^2=d^12=1,e^2=c,a*b=b*a,c*a*c=e*b*e^-1=a^-1,a*d=d*a,e*a*e^-1=b,c*b*c=d*b*d^-1=b^-1,c*d=d*c,c*e=e*c,e*d*e^-1=c*d^-1>;
// generators/relations

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