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G = C3⋊D12⋊S3  order 432 = 24·33

2nd semidirect product of C3⋊D12 and S3 acting via S3/C3=C2

metabelian, supersoluble, monomial

Aliases: D6.4(S32), C3⋊D12⋊2S3, D6⋊S3⋊4S3, (S3×Dic3)⋊2S3, (S3×C6).10D6, C33⋊3(C4○D4), C33⋊6D4⋊4C2, C33⋊4Q8⋊4C2, Dic3.2(S32), C3⋊Dic3.19D6, C3⋊2(D12⋊5S3), C3⋊3(D6.3D6), C3⋊1(D6.4D6), (C3×Dic3).21D6, C32⋊16(C4○D12), C32⋊9(D4⋊2S3), (C32×C6).15C23, C33⋊5C4.4C22, (C32×Dic3).4C22, C2.15(S33), C6.15(C2×S32), (C3×S3×Dic3)⋊4C2, (S3×C3⋊Dic3)⋊8C2, (C2×C3⋊S3).17D6, (S3×C3×C6).6C22, (C3×D6⋊S3)⋊6C2, (C3×C3⋊D12)⋊6C2, C33⋊9(C2×C4)⋊2C2, (C6×C3⋊S3).20C22, (C3×C6).64(C22×S3), (C3×C3⋊Dic3).30C22, SmallGroup(432,608)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32×C6 — C3⋊D12⋊S3
Chief series
C1 — C3 — C32 — C33 — C32×C6 — S3×C3×C6 — C3×S3×Dic3 — C3⋊D12⋊S3
Lower central
C33 — C32×C6 — C3⋊D12⋊S3
Upper central
C1 — C2

Generators and relations for C3⋊D12⋊S3
 G = < a,b,c,d,e | a3=b12=c2=d3=e2=1, bab-1=cac=a-1, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece=b6c, ede=d-1 >

Subgroups: 1116 in 210 conjugacy classes, 46 normal (all characteristic)
C1, C2, C2 [×3], C3 [×3], C3 [×4], C4 [×4], C22 [×3], S3 [×5], C6 [×3], C6 [×12], C2×C4 [×3], D4 [×3], Q8, C32 [×3], C32 [×4], Dic3, Dic3 [×13], C12 [×5], D6 [×2], D6 [×3], C2×C6 [×7], C4○D4, C3×S3 [×10], C3⋊S3, C3×C6 [×3], C3×C6 [×6], Dic6 [×4], C4×S3 [×4], D12, C2×Dic3 [×5], C3⋊D4 [×7], C2×C12, C3×D4 [×2], C33, C3×Dic3 [×2], C3×Dic3 [×7], C3⋊Dic3 [×2], C3⋊Dic3 [×7], C3×C12, S3×C6 [×4], S3×C6 [×5], C2×C3⋊S3, C62 [×2], C4○D12, D4⋊2S3 [×2], S3×C32 [×2], C3×C3⋊S3, C32×C6, S3×Dic3, S3×Dic3 [×5], C6.D6, D6⋊S3, D6⋊S3 [×3], C3⋊D12, C32⋊2Q8 [×3], S3×C12, C3×D12, C6×Dic3, C3×C3⋊D4 [×3], C32⋊4Q8, C2×C3⋊Dic3, C32⋊7D4, C32×Dic3, C3×C3⋊Dic3 [×2], C33⋊5C4, S3×C3×C6 [×2], C6×C3⋊S3, D12⋊5S3, D6.3D6, D6.4D6, C3×S3×Dic3, C3×D6⋊S3, C3×C3⋊D12, S3×C3⋊Dic3, C33⋊6D4, C33⋊4Q8, C33⋊9(C2×C4), C3⋊D12⋊S3
Quotients: C1, C2 [×7], C22 [×7], S3 [×3], C23, D6 [×9], C4○D4, C22×S3 [×3], S32 [×3], C4○D12, D4⋊2S3 [×2], C2×S32 [×3], D12⋊5S3, D6.3D6, D6.4D6, S33, C3⋊D12⋊S3

Smallest permutation representation of C3⋊D12⋊S3
►On 48 points
Generators in S48
(1 9 5)(2 6 10)(3 11 7)(4 8 12)(13 17 21)(14 22 18)(15 19 23)(16 24 20)(25 33 29)(26 30 34)(27 35 31)(28 32 36)(37 41 45)(38 46 42)(39 43 47)(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 27)(2 26)(3 25)(4 36)(5 35)(6 34)(7 33)(8 32)(9 31)(10 30)(11 29)(12 28)(13 45)(14 44)(15 43)(16 42)(17 41)(18 40)(19 39)(20 38)(21 37)(22 48)(23 47)(24 46)

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

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

42 conjugacy classes

class 1 2A2B2C2D3A3B3C3D3E3F3G4A4B4C4D4E6A6B6C6D6E6F6G6H6I6J···6P6Q12A12B12C12D12E12F12G12H
order122223333333444446666666666···661212121212121212
size1166182224448699185422244466812···123666121212181836

42 irreducible representations

dim11111111222222222444444488
type+++++++++++++++++-+--+-
imageC1C2C2C2C2C2C2C2S3S3S3D6D6D6D6C4○D4C4○D12S32S32D4⋊2S3C2×S32D12⋊5S3D6.3D6D6.4D6S33C3⋊D12⋊S3
kernelC3⋊D12⋊S3C3×S3×Dic3C3×D6⋊S3C3×C3⋊D12S3×C3⋊Dic3C33⋊6D4C33⋊4Q8C33⋊9(C2×C4)S3×Dic3D6⋊S3C3⋊D12C3×Dic3C3⋊Dic3S3×C6C2×C3⋊S3C33C32Dic3D6C32C6C3C3C3C2C1
# reps11111111111224124122322211

Matrix representation of C3⋊D12⋊S3 ►in GL8(𝔽13)

10000000
01000000
00100000
00010000
000012100
000012000
00000010
00000001
,
012000000
10000000
001200000
000120000
00000100
00001000
00000001
000000121
,
10000000
012000000
001200000
000120000
000001200
000012000
000000121
00000001
,
10000000
01000000
00010000
0012120000
00001000
00000100
00000010
00000001
,
08000000
50000000
001200000
00110000
000012000
000001200
000000120
000000012

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

C3⋊D12⋊S3 in GAP, Magma, Sage, TeX 

C_3\rtimes D_{12}\rtimes S_3
 
% in TeX
 
G:=Group("C3:D12:S3");
 
// GroupNames label
 
G:=SmallGroup(432,608);
 
// by ID
 
G=gap.SmallGroup(432,608);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,64,254,135,298,2028,14118]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^3=b^12=c^2=d^3=e^2=1,b*a*b^-1=c*a*c=a^-1,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e=b^6*c,e*d*e=d^-1>;
 
// generators/relations
 

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