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G = D6.6S32  order 432 = 24·33

3rd non-split extension by D6 of S32 acting via S32/C3⋊S3=C2

metabelian, supersoluble, monomial

Aliases: D6.6S32, Dic3.7S32, C3⋊D12⋊6S3, (S3×C6).12D6, C32⋊2Q8⋊5S3, C33⋊6(C4○D4), C33⋊9D4⋊9C2, C3⋊Dic3.32D6, C3⋊2(D12⋊S3), C3⋊3(D6.4D6), (C3×Dic3).12D6, (C32×C6).18C23, C33⋊5C4.6C22, C32⋊11(D4⋊2S3), C32⋊10(Q8⋊3S3), (C32×Dic3).6C22, C2.18S33, C6.18(C2×S32), (S3×C3⋊Dic3)⋊3C2, (Dic3×C3⋊S3)⋊2C2, (C2×C3⋊S3).33D6, (S3×C3×C6).9C22, (C3×C3⋊D12)⋊8C2, (C3×C32⋊2Q8)⋊6C2, (C6×C3⋊S3).22C22, (C3×C6).67(C22×S3), (C3×C3⋊Dic3).15C22, SmallGroup(432,611)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32×C6 — D6.6S32
Chief series
C1 — C3 — C32 — C33 — C32×C6 — S3×C3×C6 — C3×C3⋊D12 — D6.6S32
Lower central
C33 — C32×C6 — D6.6S32
Upper central
C1 — C2

Generators and relations for D6.6S32
 G = < a,b,c,d,e | a6=b3=c2=1, d6=e2=a3, ab=ba, ac=ca, dad-1=eae-1=a-1, cbc=b-1, bd=db, be=eb, dcd-1=a3c, ce=ec, ede-1=a3d5 >

Subgroups: 1156 in 214 conjugacy classes, 46 normal (18 characteristic)
C1, C2, C2, C3, C3, C3, C4, C22, S3, C6, C6, C6, C2×C4, D4, Q8, C32, C32, C32, Dic3, Dic3, C12, D6, D6, C2×C6, C4○D4, C3×S3, C3⋊S3, C3×C6, C3×C6, C3×C6, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C3×D4, C3×Q8, C33, C3×Dic3, C3×Dic3, C3⋊Dic3, C3⋊Dic3, C3×C12, S3×C6, S3×C6, C2×C3⋊S3, C62, D4⋊2S3, Q8⋊3S3, S3×C32, C3×C3⋊S3, C32×C6, S3×Dic3, D6⋊S3, C3⋊D12, C3⋊D12, C32⋊2Q8, C3×Dic6, C3×D12, C3×C3⋊D4, C4×C3⋊S3, C2×C3⋊Dic3, C32×Dic3, C3×C3⋊Dic3, C33⋊5C4, S3×C3×C6, C6×C3⋊S3, D12⋊S3, D6.4D6, C3×C3⋊D12, C3×C32⋊2Q8, S3×C3⋊Dic3, Dic3×C3⋊S3, C33⋊9D4, D6.6S32
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C22×S3, S32, D4⋊2S3, Q8⋊3S3, C2×S32, D12⋊S3, D6.4D6, S33, D6.6S32

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

39 conjugacy classes

class 1 2A2B2C2D3A3B3C3D3E3F3G4A4B4C4D4E6A6B6C6D6E6F6G6H6I6J6K6L6M12A···12H12I
order12222333333344444666666666666612···1212
size1161818222444866182727222444812121212363612···1236

39 irreducible representations

dim1111112222222444444488
type++++++++++++++-++-+-
imageC1C2C2C2C2C2S3S3D6D6D6D6C4○D4S32S32D4⋊2S3Q8⋊3S3C2×S32D12⋊S3D6.4D6S33D6.6S32
kernelD6.6S32C3×C3⋊D12C3×C32⋊2Q8S3×C3⋊Dic3Dic3×C3⋊S3C33⋊9D4C3⋊D12C32⋊2Q8C3×Dic3C3⋊Dic3S3×C6C2×C3⋊S3C33Dic3D6C32C32C6C3C3C2C1
# reps1211212141222212134211

Matrix representation of D6.6S32 ►in GL8(𝔽13)

120000000
012000000
00100000
00010000
0000121200
00001000
00000010
00000001
,
10000000
01000000
00100000
00010000
00001000
00000100
000000012
000000112
,
116000000
62000000
00100000
00010000
00001000
00000100
000000012
000000120
,
01000000
120000000
001120000
00100000
000012000
00001100
00000010
00000001
,
34000000
410000000
001200000
001210000
00001000
0000121200
000000120
000000012

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

D6.6S32 in GAP, Magma, Sage, TeX 

D_6._6S_3^2
 
% in TeX
 
G:=Group("D6.6S3^2");
 
// GroupNames label
 
G:=SmallGroup(432,611);
 
// by ID
 
G=gap.SmallGroup(432,611);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,141,64,254,135,58,298,2028,14118]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^6=b^3=c^2=1,d^6=e^2=a^3,a*b=b*a,a*c=c*a,d*a*d^-1=e*a*e^-1=a^-1,c*b*c=b^-1,b*d=d*b,b*e=e*b,d*c*d^-1=a^3*c,c*e=e*c,e*d*e^-1=a^3*d^5>;
 
// generators/relations
 

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