Copied to
clipboard

G = C12⋊S3⋊12S3  order 432 = 24·33

6th semidirect product of C12⋊S3 and S3 acting via S3/C3=C2

metabelian, supersoluble, monomial

Aliases: C12.43S32, C12⋊S3⋊12S3, (C3×C12).148D6, C33⋊19(C4○D4), C33⋊9D4⋊11C2, C3⋊Dic3.21D6, C3⋊5(D12⋊S3), C3⋊5(D12⋊5S3), C32⋊4Q8⋊13S3, C3⋊5(D6.6D6), C32⋊13(C4○D12), C32⋊8(Q8⋊3S3), C4.5(C32⋊4D6), (C32×C6).66C23, C32⋊14(D4⋊2S3), (C32×C12).50C22, (C4×C3⋊S3)⋊9S3, C6.95(C2×S32), (C12×C3⋊S3)⋊8C2, C33⋊9(C2×C4)⋊4C2, (C2×C3⋊S3).46D6, (C3×C12⋊S3)⋊12C2, (C6×C3⋊S3).30C22, (C3×C32⋊4Q8)⋊13C2, C2.4(C2×C32⋊4D6), (C3×C6).116(C22×S3), (C3×C3⋊Dic3).24C22, SmallGroup(432,688)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32×C6 — C12⋊S3⋊12S3
Chief series
C1 — C3 — C32 — C33 — C32×C6 — C6×C3⋊S3 — C33⋊9(C2×C4) — C12⋊S3⋊12S3
Lower central
C33 — C32×C6 — C12⋊S3⋊12S3
Upper central
C1 — C2 — C4

Generators and relations for C12⋊S3⋊12S3
 G = < a,b,c,d | a3=b12=1, c6=d2=b6, ab=ba, cac-1=a-1, ad=da, cbc-1=b5, dbd-1=b-1, dcd-1=c5 >

Subgroups: 1064 in 210 conjugacy classes, 47 normal (35 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C2×C4, D4, Q8, C32, C32, Dic3, C12, C12, D6, C2×C6, C4○D4, C3×S3, C3⋊S3, C3×C6, C3×C6, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C3×Q8, C33, C3×Dic3, C3⋊Dic3, C3⋊Dic3, C3×C12, C3×C12, S3×C6, C2×C3⋊S3, C2×C3⋊S3, C4○D12, D4⋊2S3, Q8⋊3S3, C3×C3⋊S3, C32×C6, S3×Dic3, C6.D6, D6⋊S3, C3⋊D12, C3×Dic6, S3×C12, C3×D12, C32⋊4Q8, C4×C3⋊S3, C12⋊S3, C3×C3⋊Dic3, C3×C3⋊Dic3, C32×C12, C6×C3⋊S3, C6×C3⋊S3, D12⋊5S3, D12⋊S3, D6.6D6, C33⋊9(C2×C4), C33⋊9D4, C3×C32⋊4Q8, C12×C3⋊S3, C3×C12⋊S3, C12⋊S3⋊12S3
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C22×S3, S32, C4○D12, D4⋊2S3, Q8⋊3S3, C2×S32, C32⋊4D6, D12⋊5S3, D12⋊S3, D6.6D6, C2×C32⋊4D6, C12⋊S3⋊12S3

Smallest permutation representation of C12⋊S3⋊12S3
►On 48 points
Generators in S48
(1 5 9)(2 6 10)(3 7 11)(4 8 12)(13 17 21)(14 18 22)(15 19 23)(16 20 24)(25 33 29)(26 34 30)(27 35 31)(28 36 32)(37 45 41)(38 46 42)(39 47 43)(40 48 44)

(1 2 3 4 5 6 7 8 9 10 11 12)(13 14 15 16 17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32 33 34 35 36)(37 38 39 40 41 42 43 44 45 46 47 48)

(1 13 3 23 5 21 7 19 9 17 11 15)(2 18 4 16 6 14 8 24 10 22 12 20)(25 38 35 40 33 42 31 44 29 46 27 48)(26 43 36 45 34 47 32 37 30 39 28 41)

(1 45 7 39)(2 44 8 38)(3 43 9 37)(4 42 10 48)(5 41 11 47)(6 40 12 46)(13 30 19 36)(14 29 20 35)(15 28 21 34)(16 27 22 33)(17 26 23 32)(18 25 24 31)
 
G:=sub<Sym(48)| (1,5,9)(2,6,10)(3,7,11)(4,8,12)(13,17,21)(14,18,22)(15,19,23)(16,20,24)(25,33,29)(26,34,30)(27,35,31)(28,36,32)(37,45,41)(38,46,42)(39,47,43)(40,48,44), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48), (1,13,3,23,5,21,7,19,9,17,11,15)(2,18,4,16,6,14,8,24,10,22,12,20)(25,38,35,40,33,42,31,44,29,46,27,48)(26,43,36,45,34,47,32,37,30,39,28,41), (1,45,7,39)(2,44,8,38)(3,43,9,37)(4,42,10,48)(5,41,11,47)(6,40,12,46)(13,30,19,36)(14,29,20,35)(15,28,21,34)(16,27,22,33)(17,26,23,32)(18,25,24,31)>;
 
G:=Group( (1,5,9)(2,6,10)(3,7,11)(4,8,12)(13,17,21)(14,18,22)(15,19,23)(16,20,24)(25,33,29)(26,34,30)(27,35,31)(28,36,32)(37,45,41)(38,46,42)(39,47,43)(40,48,44), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48), (1,13,3,23,5,21,7,19,9,17,11,15)(2,18,4,16,6,14,8,24,10,22,12,20)(25,38,35,40,33,42,31,44,29,46,27,48)(26,43,36,45,34,47,32,37,30,39,28,41), (1,45,7,39)(2,44,8,38)(3,43,9,37)(4,42,10,48)(5,41,11,47)(6,40,12,46)(13,30,19,36)(14,29,20,35)(15,28,21,34)(16,27,22,33)(17,26,23,32)(18,25,24,31) );
 
G=PermutationGroup([[(1,5,9),(2,6,10),(3,7,11),(4,8,12),(13,17,21),(14,18,22),(15,19,23),(16,20,24),(25,33,29),(26,34,30),(27,35,31),(28,36,32),(37,45,41),(38,46,42),(39,47,43),(40,48,44)], [(1,2,3,4,5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32,33,34,35,36),(37,38,39,40,41,42,43,44,45,46,47,48)], [(1,13,3,23,5,21,7,19,9,17,11,15),(2,18,4,16,6,14,8,24,10,22,12,20),(25,38,35,40,33,42,31,44,29,46,27,48),(26,43,36,45,34,47,32,37,30,39,28,41)], [(1,45,7,39),(2,44,8,38),(3,43,9,37),(4,42,10,48),(5,41,11,47),(6,40,12,46),(13,30,19,36),(14,29,20,35),(15,28,21,34),(16,27,22,33),(17,26,23,32),(18,25,24,31)]])
 

48 conjugacy classes

class 1 2A2B2C2D3A3B3C3D···3H4A4B4C4D4E6A6B6C6D···6H6I6J6K6L12A12B12C···12N12O12P12Q12R
order122223333···3444446666···66666121212···1212121212
size111818182224···429918182224···418183636224···418183636

48 irreducible representations

dim111111222222224444444444
type+++++++++++++-++-+
imageC1C2C2C2C2C2S3S3S3D6D6D6C4○D4C4○D12S32D4⋊2S3Q8⋊3S3C2×S32C32⋊4D6D12⋊5S3D12⋊S3D6.6D6C2×C32⋊4D6C12⋊S3⋊12S3
kernelC12⋊S3⋊12S3C33⋊9(C2×C4)C33⋊9D4C3×C32⋊4Q8C12×C3⋊S3C3×C12⋊S3C32⋊4Q8C4×C3⋊S3C12⋊S3C3⋊Dic3C3×C12C2×C3⋊S3C33C32C12C32C32C6C4C3C3C3C2C1
# reps122111111333243113222224

Matrix representation of C12⋊S3⋊12S3 ►in GL6(𝔽13)

100000
010000
00121200
001000
000010
000001
,
010000
1210000
0012000
0001200
000037
0000610
,
010000
100000
001000
00121200
000085
000080
,
0120000
1200000
001000
000100
000005
000050

G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,1,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,12,0,0,0,0,1,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,3,6,0,0,0,0,7,10],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,12,0,0,0,0,0,12,0,0,0,0,0,0,8,8,0,0,0,0,5,0],[0,12,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,5,0,0,0,0,5,0] >;
 

C12⋊S3⋊12S3 in GAP, Magma, Sage, TeX 

C_{12}\rtimes S_3\rtimes_{12}S_3
 
% in TeX
 
G:=Group("C12:S3:12S3");
 
// GroupNames label
 
G:=SmallGroup(432,688);
 
// by ID
 
G=gap.SmallGroup(432,688);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,64,254,58,1124,571,2028,14118]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^3=b^12=1,c^6=d^2=b^6,a*b=b*a,c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=b^5,d*b*d^-1=b^-1,d*c*d^-1=c^5>;
 
// generators/relations
 

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁