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G = D4×C32⋊C6  order 432 = 24·33

Direct product of D4 and C32⋊C6

direct product, metabelian, supersoluble, monomial

Aliases: D4×C32⋊C6, C62⋊3D6, (C3×C12)⋊3D6, (D4×He3)⋊3C2, He3⋊10(C2×D4), C62⋊1(C2×C6), C12⋊S3⋊3C6, C32⋊3(C6×D4), C32⋊6(S3×D4), C12.15(S3×C6), He3⋊4D4⋊8C2, He3⋊6D4⋊5C2, (D4×C32)⋊2C6, (D4×C32)⋊3S3, C32⋊7D4⋊1C6, (C4×He3)⋊3C22, C32⋊C12⋊8C22, (C2×He3).23C23, (C22×He3)⋊3C22, (D4×C3⋊S3)⋊C3, (C3×C12)⋊(C2×C6), (C4×C3⋊S3)⋊1C6, C3.2(C3×S3×D4), C3⋊S3⋊2(C3×D4), C6.33(S3×C2×C6), C4⋊1(C2×C32⋊C6), (C22×C3⋊S3)⋊2C6, (C4×C32⋊C6)⋊5C2, (C2×C6).10(S3×C6), C3⋊Dic3⋊1(C2×C6), (C3×D4).9(C3×S3), C22⋊3(C2×C32⋊C6), (C3×C6).5(C22×C6), (C22×C32⋊C6)⋊5C2, (C2×C32⋊C6)⋊8C22, (C3×C6).27(C22×S3), C2.6(C22×C32⋊C6), (C2×C3⋊S3)⋊2(C2×C6), SmallGroup(432,360)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3×C6 — D4×C32⋊C6
Chief series
C1 — C3 — C32 — C3×C6 — C2×He3 — C2×C32⋊C6 — C22×C32⋊C6 — D4×C32⋊C6
Lower central
C32 — C3×C6 — D4×C32⋊C6
Upper central
C1 — C2 — D4

Generators and relations for D4×C32⋊C6
 G = < a,b,c,d,e | a4=b2=c3=d3=e6=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=c-1d-1, ede-1=d-1 >

Subgroups: 1065 in 205 conjugacy classes, 56 normal (30 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, D4, D4, C23, C32, C32, Dic3, C12, C12, D6, C2×C6, C2×C6, C2×D4, C3×S3, C3⋊S3, C3⋊S3, C3×C6, C3×C6, C4×S3, D12, C3⋊D4, C2×C12, C3×D4, C3×D4, C22×S3, C22×C6, He3, C3×Dic3, C3⋊Dic3, C3×C12, C3×C12, S3×C6, C2×C3⋊S3, C2×C3⋊S3, C2×C3⋊S3, C62, C62, S3×D4, C6×D4, C32⋊C6, C32⋊C6, C2×He3, C2×He3, S3×C12, C3×D12, C3×C3⋊D4, C4×C3⋊S3, C12⋊S3, C32⋊7D4, D4×C32, D4×C32, S3×C2×C6, C22×C3⋊S3, C32⋊C12, C4×He3, C2×C32⋊C6, C2×C32⋊C6, C2×C32⋊C6, C22×He3, C3×S3×D4, D4×C3⋊S3, C4×C32⋊C6, He3⋊4D4, He3⋊6D4, D4×He3, C22×C32⋊C6, D4×C32⋊C6
Quotients: C1, C2, C3, C22, S3, C6, D4, C23, D6, C2×C6, C2×D4, C3×S3, C3×D4, C22×S3, C22×C6, S3×C6, S3×D4, C6×D4, C32⋊C6, S3×C2×C6, C2×C32⋊C6, C3×S3×D4, C22×C32⋊C6, D4×C32⋊C6

Smallest permutation representation of D4×C32⋊C6
►On 36 points
Generators in S36
(1 4 9 12)(2 5 7 10)(3 6 8 11)(13 26 22 34)(14 27 23 35)(15 28 24 36)(16 29 19 31)(17 30 20 32)(18 25 21 33)

(1 9)(2 7)(3 8)(25 33)(26 34)(27 35)(28 36)(29 31)(30 32)

(1 30 27)(3 29 26)(4 20 23)(6 19 22)(8 31 34)(9 32 35)(11 16 13)(12 17 14)

(1 27 30)(2 25 28)(3 29 26)(4 23 20)(5 21 24)(6 19 22)(7 33 36)(8 31 34)(9 35 32)(10 18 15)(11 16 13)(12 14 17)

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

50 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B3C3D3E3F4A4B6A6B6C6D6E6F···6L6M6N6O6P6Q···6V6W6X6Y6Z12A12B12C12D12E12F12G12H
order1222222233333344666666···666666···666661212121212121212
size1122991818233666218233446···6999912···12181818184661212121818

50 irreducible representations

dim111111111111122222222244666
type+++++++++++++++
imageC1C2C2C2C2C2C3C6C6C6C6C6D4×C32⋊C6S3D4D6D6C3×S3C3×D4S3×C6S3×C6S3×D4C3×S3×D4C32⋊C6C2×C32⋊C6C2×C32⋊C6
kernelD4×C32⋊C6C4×C32⋊C6He3⋊4D4He3⋊6D4D4×He3C22×C32⋊C6D4×C3⋊S3C4×C3⋊S3C12⋊S3C32⋊7D4D4×C32C22×C3⋊S3C1D4×C32C32⋊C6C3×C12C62C3×D4C3⋊S3C12C2×C6C32C3D4C4C22
# reps11121222242411212242412112

Matrix representation of D4×C32⋊C6 ►in GL10(𝔽13)

0010000000
0001000000
12000000000
01200000000
00001200000
00000120000
00000012000
00000001200
00000000120
00000000012
,
1000000000
0100000000
00120000000
00012000000
0000100000
0000010000
0000001000
0000000100
0000000010
0000000001
,
01200000000
11200000000
00012000000
00112000000
00001210000
00001200000
00000001200
00000011200
0000900310
0000900301
,
1000000000
0100000000
0010000000
0001000000
00000120000
00001120000
00000001200
00000011200
0000401001212
0000090310
,
11200000000
01200000000
00112000000
00012000000
0000000100
0000001000
00004410101112
00000000121
0000000030
0000010030

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

D4×C32⋊C6 in GAP, Magma, Sage, TeX 

D_4\times C_3^2\rtimes C_6
 
% in TeX
 
G:=Group("D4xC3^2:C6");
 
// GroupNames label
 
G:=SmallGroup(432,360);
 
// by ID
 
G=gap.SmallGroup(432,360);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,303,4037,1034,14118]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^4=b^2=c^3=d^3=e^6=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=c^-1*d^-1,e*d*e^-1=d^-1>;
 
// generators/relations
 

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