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G = C3⋊S3⋊3D12  order 432 = 24·33

The semidirect product of C3⋊S3 and D12 acting via D12/C12=C2

metabelian, supersoluble, monomial

Aliases: C12⋊3(S32), C3⋊S3⋊3D12, C3⋊4(S3×D12), (C3×C12)⋊14D6, C33⋊16(C2×D4), C3⋊Dic3⋊20D6, C4⋊(C32⋊4D6), C12⋊S3⋊13S3, C32⋊12(S3×D4), C32⋊7(C2×D12), C33⋊9D4⋊4C2, C3⋊3(D6⋊D6), (C32×C12)⋊6C22, (C32×C6).69C23, C6.98(C2×S32), (C3×C3⋊S3)⋊9D4, (C4×C3⋊S3)⋊10S3, (C12×C3⋊S3)⋊9C2, (C2×C3⋊S3)⋊11D6, (C6×C3⋊S3)⋊12C22, (C3×C12⋊S3)⋊13C2, (C2×C32⋊4D6)⋊3C2, C2.6(C2×C32⋊4D6), (C3×C6).119(C22×S3), (C3×C3⋊Dic3)⋊15C22, SmallGroup(432,691)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32×C6 — C3⋊S3⋊3D12
Chief series
C1 — C3 — C32 — C33 — C32×C6 — C6×C3⋊S3 — C2×C32⋊4D6 — C3⋊S3⋊3D12
Lower central
C33 — C32×C6 — C3⋊S3⋊3D12
Upper central
C1 — C2 — C4

Generators and relations for C3⋊S3⋊3D12
 G = < a,b,c,d,e | a3=b3=c2=d12=e2=1, ab=ba, cac=a-1, ad=da, ae=ea, cbc=ebe=b-1, bd=db, cd=dc, ce=ec, ede=d-1 >

Subgroups: 1752 in 270 conjugacy classes, 51 normal (19 characteristic)
C1, C2, C2 [×6], C3, C3 [×2], C3 [×4], C4, C4, C22 [×9], S3 [×18], C6, C6 [×2], C6 [×10], C2×C4, D4 [×4], C23 [×2], C32, C32 [×2], C32 [×4], Dic3 [×3], C12, C12 [×2], C12 [×5], D6 [×27], C2×C6 [×5], C2×D4, C3×S3 [×18], C3⋊S3 [×2], C3⋊S3 [×4], C3×C6, C3×C6 [×2], C3×C6 [×4], C4×S3 [×3], D12 [×8], C3⋊D4 [×4], C2×C12, C3×D4 [×2], C22×S3 [×6], C33, C3×Dic3 [×3], C3⋊Dic3, C3×C12, C3×C12 [×2], C3×C12 [×4], S32 [×12], S3×C6 [×15], C2×C3⋊S3, C2×C3⋊S3 [×4], C2×D12, S3×D4 [×2], C3×C3⋊S3 [×2], C3×C3⋊S3 [×4], C32×C6, D6⋊S3 [×2], C3⋊D12 [×4], S3×C12 [×3], C3×D12 [×6], C4×C3⋊S3, C12⋊S3 [×2], C2×S32 [×6], C3×C3⋊Dic3, C32×C12, C32⋊4D6 [×4], C6×C3⋊S3, C6×C3⋊S3 [×4], S3×D12 [×2], D6⋊D6, C33⋊9D4 [×2], C12×C3⋊S3, C3×C12⋊S3 [×2], C2×C32⋊4D6 [×2], C3⋊S3⋊3D12
Quotients: C1, C2 [×7], C22 [×7], S3 [×3], D4 [×2], C23, D6 [×9], C2×D4, D12 [×2], C22×S3 [×3], S32 [×3], C2×D12, S3×D4 [×2], C2×S32 [×3], C32⋊4D6, S3×D12 [×2], D6⋊D6, C2×C32⋊4D6, C3⋊S3⋊3D12

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

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

48 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B3C3D···3H4A4B6A6B6C6D···6H6I6J6K6L6M6N12A12B12C···12N12O12P
order122222223333···3446666···6666666121212···121212
size1199181818182224···42182224···4181836363636224···41818

48 irreducible representations

dim11111222222244444444
type++++++++++++++++
imageC1C2C2C2C2S3S3D4D6D6D6D12S32S3×D4C2×S32C32⋊4D6S3×D12D6⋊D6C2×C32⋊4D6C3⋊S3⋊3D12
kernelC3⋊S3⋊3D12C33⋊9D4C12×C3⋊S3C3×C12⋊S3C2×C32⋊4D6C4×C3⋊S3C12⋊S3C3×C3⋊S3C3⋊Dic3C3×C12C2×C3⋊S3C3⋊S3C12C32C6C4C3C3C2C1
# reps12122122135432324224

Matrix representation of C3⋊S3⋊3D12 ►in GL8(𝔽13)

10000000
01000000
00100000
00010000
00001000
00000100
000000012
000000112
,
10000000
01000000
00100000
00010000
00000100
0000121200
00000010
00000001
,
120000000
012000000
001200000
000120000
00001000
0000121200
00000001
00000010
,
1010000000
123000000
000120000
00110000
00001000
00000100
00000010
00000001
,
06000000
110000000
00010000
00100000
000012000
00001100
000000120
000000012

G:=sub<GL(8,GF(13))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[10,12,0,0,0,0,0,0,10,3,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,11,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12] >;
 

C3⋊S3⋊3D12 in GAP, Magma, Sage, TeX 

C_3\rtimes S_3\rtimes_3D_{12}
 
% in TeX
 
G:=Group("C3:S3:3D12");
 
// GroupNames label
 
G:=SmallGroup(432,691);
 
// by ID
 
G=gap.SmallGroup(432,691);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,135,58,1124,571,2028,14118]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^3=b^3=c^2=d^12=e^2=1,a*b=b*a,c*a*c=a^-1,a*d=d*a,a*e=e*a,c*b*c=e*b*e=b^-1,b*d=d*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
 
// generators/relations
 

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