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G = D6:S3:S3order 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, (S3xC6).11D6, C33:5(C4oD4), C33:5Q8:6C2, C3:Dic3.31D6, C3:2(D6.4D6), (C32xC6).17C23, C33:5C4.5C22, C32:15(D4:2S3), C2.17S33, C6.17(C2xS32), (S3xC3:Dic3):2C2, (S3xC3xC6).8C22, (C3xD6:S3):7C2, (C3xC6).66(C22xS3), (C3xC3:Dic3).14C22, SmallGroup(432,610)

Series: Derived Chief Lower central Upper central

C1C32xC6 — D6:S3:S3
C1C3C32C33C32xC6S3xC3xC6C3xD6:S3 — D6:S3:S3
C33C32xC6 — D6:S3:S3
C1C2

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, C2xC4, D4, Q8, C32, C32, Dic3, C12, D6, C2xC6, C4oD4, C3xS3, C3xC6, C3xC6, Dic6, C4xS3, C2xDic3, C3:D4, C3xD4, C33, C3xDic3, C3:Dic3, C3:Dic3, S3xC6, S3xC6, C62, D4:2S3, S3xC32, C32xC6, S3xDic3, D6:S3, C32:2Q8, C3xC3:D4, C2xC3:Dic3, C3xC3:Dic3, C33:5C4, S3xC3xC6, D6.4D6, C3xD6:S3, S3xC3:Dic3, C33:5Q8, D6:S3:S3
Quotients: C1, C2, C22, S3, C23, D6, C4oD4, C22xS3, S32, D4:2S3, C2xS32, 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++++++++-+-+-
imageC1C2C2C2S3D6D6C4oD4S32D4:2S3C2xS32D6.4D6S33D6:S3:S3
kernelD6:S3:S3C3xD6:S3S3xC3:Dic3C33:5Q8D6:S3C3:Dic3S3xC6C33D6C32C6C3C2C1
# 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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