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G = D6.S32  order 432 = 24·33

2nd non-split extension by D6 of S32 acting via S32/C3×S3=C2

metabelian, supersoluble, monomial

Aliases: D6.2S32, Dic3.9S32, C3⋊D12⋊7S3, (S3×Dic3)⋊5S3, (S3×C6).20D6, C6.D6⋊2S3, C33⋊2(C4○D4), C33⋊9D4⋊3C2, C33⋊6D4⋊3C2, C33⋊4Q8⋊3C2, C3⋊Dic3.18D6, C3⋊1(D12⋊5S3), C3⋊2(D6.D6), C3⋊2(D6.3D6), C32⋊7(C4○D12), (C3×Dic3).10D6, C32⋊8(D4⋊2S3), (C32×C6).14C23, C33⋊5C4.3C22, (C32×Dic3).20C22, C2.14S33, C6.14(C2×S32), (C3×S3×Dic3)⋊2C2, (Dic3×C3⋊S3)⋊8C2, (C2×C3⋊S3).32D6, (S3×C3×C6).5C22, (C3×C3⋊D12)⋊5C2, (C3×C6.D6)⋊2C2, (C6×C3⋊S3).19C22, (C3×C6).63(C22×S3), (C3×C3⋊Dic3).6C22, SmallGroup(432,607)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32×C6 — D6.S32
Chief series
C1 — C3 — C32 — C33 — C32×C6 — S3×C3×C6 — C3×S3×Dic3 — D6.S32
Lower central
C33 — C32×C6 — D6.S32
Upper central
C1 — C2

Generators and relations for D6.S32
 G = < a,b,c,d,e | a3=b12=c2=d3=e2=1, bab-1=cac=eae=a-1, ad=da, cbc=b-1, bd=db, be=eb, cd=dc, ece=b6c, ede=d-1 >

Subgroups: 1156 in 210 conjugacy classes, 46 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, C2×C4, D4, Q8, C32, C32, Dic3, Dic3, C12, D6, 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, C33, C3×Dic3, C3×Dic3, C3⋊Dic3, C3⋊Dic3, C3×C12, S3×C6, S3×C6, C2×C3⋊S3, C62, C4○D12, D4⋊2S3, S3×C32, C3×C3⋊S3, C32×C6, S3×Dic3, S3×Dic3, C6.D6, D6⋊S3, C3⋊D12, C3⋊D12, C32⋊2Q8, S3×C12, C3×D12, C6×Dic3, C3×C3⋊D4, C32⋊4Q8, C4×C3⋊S3, C32⋊7D4, C32×Dic3, C3×C3⋊Dic3, C33⋊5C4, S3×C3×C6, C6×C3⋊S3, D12⋊5S3, D6.D6, D6.3D6, C3×S3×Dic3, C3×C6.D6, C3×C3⋊D12, Dic3×C3⋊S3, C33⋊6D4, C33⋊4Q8, C33⋊9D4, D6.S32
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C22×S3, S32, C4○D12, D4⋊2S3, C2×S32, D12⋊5S3, D6.D6, D6.3D6, S33, D6.S32

Smallest permutation representation of D6.S32
►On 48 points
Generators in S48
(1 5 9)(2 10 6)(3 7 11)(4 12 8)(13 17 21)(14 22 18)(15 19 23)(16 24 20)(25 29 33)(26 34 30)(27 31 35)(28 36 32)(37 45 41)(38 42 46)(39 47 43)(40 44 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 33)(2 32)(3 31)(4 30)(5 29)(6 28)(7 27)(8 26)(9 25)(10 36)(11 35)(12 34)(13 40)(14 39)(15 38)(16 37)(17 48)(18 47)(19 46)(20 45)(21 44)(22 43)(23 42)(24 41)

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

45 conjugacy classes

class 1 2A2B2C2D3A3B3C3D3E3F3G4A4B4C4D4E6A6B6C6D6E6F6G6H6I6J6K6L6M6N6O12A···12F12G···12K12L12M
order1222233333334444466666666666666612···1212···121212
size1161818222444833618542224446681212121818366···612···121818

45 irreducible representations

dim11111111222222222444444488
type+++++++++++++++++-+-+-
imageC1C2C2C2C2C2C2C2S3S3S3D6D6D6D6C4○D4C4○D12S32S32D4⋊2S3C2×S32D12⋊5S3D6.D6D6.3D6S33D6.S32
kernelD6.S32C3×S3×Dic3C3×C6.D6C3×C3⋊D12Dic3×C3⋊S3C33⋊6D4C33⋊4Q8C33⋊9D4S3×Dic3C6.D6C3⋊D12C3×Dic3C3⋊Dic3S3×C6C2×C3⋊S3C33C32Dic3D6C32C6C3C3C3C2C1
# reps11111111111412228211322211

Matrix representation of D6.S32 ►in GL8(𝔽13)

10000000
01000000
00100000
00010000
00001000
00000100
00000001
0000001212
,
80000000
05000000
00110000
001200000
000012000
000001200
00000010
0000001212
,
05000000
80000000
00110000
000120000
00001000
00000100
000000120
00000011
,
10000000
01000000
00100000
00010000
000012100
000012000
00000010
00000001
,
120000000
01000000
001200000
000120000
00000100
00001000
00000010
0000001212

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

D6.S32 in GAP, Magma, Sage, TeX 

D_6.S_3^2
 
% in TeX
 
G:=Group("D6.S3^2");
 
// GroupNames label
 
G:=SmallGroup(432,607);
 
// by ID
 
G=gap.SmallGroup(432,607);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,64,254,58,298,2028,14118]);
 
// 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=e*a*e=a^-1,a*d=d*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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