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G = D6⋊S3⋊S3  order 432 = 24·33

5th semidirect product of D6⋊S3 and S3 acting via S3/C3=C2

metabelian, supersoluble, monomial

Aliases: D6.5S32, D6⋊S3⋊5S3, (S3×C6).11D6, C33⋊5(C4○D4), C33⋊5Q8⋊6C2, C3⋊Dic3.31D6, C3⋊2(D6.4D6), (C32×C6).17C23, C33⋊5C4.5C22, C32⋊15(D4⋊2S3), C2.17S33, C6.17(C2×S32), (S3×C3⋊Dic3)⋊2C2, (S3×C3×C6).8C22, (C3×D6⋊S3)⋊7C2, (C3×C6).66(C22×S3), (C3×C3⋊Dic3).14C22, SmallGroup(432,610)

Series: Derived ►Chief ►Lower central ►Upper central

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

Generators and relations for D6⋊S3⋊S3
 G = < a,b,c,d,e,f | a6=b2=c3=d2=e3=f2=1, bab=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, dbd=fbf=a3b, be=eb, dcd=c-1, ce=ec, cf=fc, de=ed, fdf=a3d, fef=e-1 >

Subgroups: 1076 in 214 conjugacy classes, 46 normal (7 characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, C2×C4, D4, Q8, C32, C32, Dic3, C12, D6, C2×C6, C4○D4, C3×S3, C3×C6, C3×C6, Dic6, C4×S3, C2×Dic3, C3⋊D4, C3×D4, C33, C3×Dic3, C3⋊Dic3, C3⋊Dic3, S3×C6, S3×C6, C62, D4⋊2S3, S3×C32, C32×C6, S3×Dic3, D6⋊S3, C32⋊2Q8, C3×C3⋊D4, C2×C3⋊Dic3, C3×C3⋊Dic3, C33⋊5C4, S3×C3×C6, D6.4D6, C3×D6⋊S3, S3×C3⋊Dic3, C33⋊5Q8, D6⋊S3⋊S3
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C22×S3, S32, D4⋊2S3, C2×S32, D6.4D6, S33, D6⋊S3⋊S3

Smallest permutation representation of D6⋊S3⋊S3
►On 48 points
Generators in S48
(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 12)(2 11)(3 10)(4 9)(5 8)(6 7)(13 19)(14 24)(15 23)(16 22)(17 21)(18 20)(25 31)(26 36)(27 35)(28 34)(29 33)(30 32)(37 46)(38 45)(39 44)(40 43)(41 48)(42 47)

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

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

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

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

39 conjugacy classes

class 1 2A2B2C2D3A3B3C3D3E3F3G4A4B4C4D4E6A6B6C6D6E6F6G6H···6S12A12B12C
order1222233333334444466666666···6121212
size1166622244481818182727222444812···12363636

39 irreducible representations

dim11112222444488
type++++++++-+-+-
imageC1C2C2C2S3D6D6C4○D4S32D4⋊2S3C2×S32D6.4D6S33D6⋊S3⋊S3
kernelD6⋊S3⋊S3C3×D6⋊S3S3×C3⋊Dic3C33⋊5Q8D6⋊S3C3⋊Dic3S3×C6C33D6C32C6C3C2C1
# reps13313362333611

Matrix representation of D6⋊S3⋊S3 ►in GL8(𝔽13)

120000000
012000000
00100000
00010000
000012100
000012000
00000010
00000001
,
012000000
120000000
00100000
00010000
000012000
000012100
000000120
000000012
,
10000000
01000000
001210000
001200000
00001000
00000100
00000010
00000001
,
29000000
411000000
000120000
001200000
000012000
000001200
00000010
00000001
,
10000000
01000000
00100000
00010000
00001000
00000100
000000121
000000120
,
73000000
106000000
001200000
000120000
000012000
000001200
00000001
00000010

G:=sub<GL(8,GF(13))| [12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[2,4,0,0,0,0,0,0,9,11,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,1,0],[7,10,0,0,0,0,0,0,3,6,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0] >;
 

D6⋊S3⋊S3 in GAP, Magma, Sage, TeX 

D_6\rtimes S_3\rtimes S_3
 
% in TeX
 
G:=Group("D6:S3:S3");
 
// GroupNames label
 
G:=SmallGroup(432,610);
 
// by ID
 
G=gap.SmallGroup(432,610);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,141,254,135,298,2028,14118]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^6=b^2=c^3=d^2=e^3=f^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,d*b*d=f*b*f=a^3*b,b*e=e*b,d*c*d=c^-1,c*e=e*c,c*f=f*c,d*e=e*d,f*d*f=a^3*d,f*e*f=e^-1>;
 
// generators/relations
 

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