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G = D4xC32:C6  order 432 = 24·33

Direct product of D4 and C32:C6

direct product, metabelian, supersoluble, monomial

Aliases: D4xC32:C6, C62:3D6, (C3xC12):3D6, (D4xHe3):3C2, He3:10(C2xD4), C62:1(C2xC6), C12:S3:3C6, C32:3(C6xD4), C32:6(S3xD4), C12.15(S3xC6), He3:4D4:8C2, He3:6D4:5C2, (D4xC32):2C6, (D4xC32):3S3, C32:7D4:1C6, (C4xHe3):3C22, C32:C12:8C22, (C2xHe3).23C23, (C22xHe3):3C22, (D4xC3:S3):C3, (C3xC12):(C2xC6), (C4xC3:S3):1C6, C3.2(C3xS3xD4), C3:S3:2(C3xD4), C6.33(S3xC2xC6), C4:1(C2xC32:C6), (C22xC3:S3):2C6, (C4xC32:C6):5C2, (C2xC6).10(S3xC6), C3:Dic3:1(C2xC6), (C3xD4).9(C3xS3), C22:3(C2xC32:C6), (C3xC6).5(C22xC6), (C22xC32:C6):5C2, (C2xC32:C6):8C22, (C3xC6).27(C22xS3), C2.6(C22xC32:C6), (C2xC3:S3):2(C2xC6), SmallGroup(432,360)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3xC6 — D4xC32:C6
Chief series
C1 — C3 — C32 — C3xC6 — C2xHe3 — C2xC32:C6 — C22xC32:C6 — D4xC32:C6
Lower central
C32 — C3xC6 — D4xC32:C6
Upper central
C1 — C2 — D4

Generators and relations for D4xC32: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, C2xC4, D4, D4, C23, C32, C32, Dic3, C12, C12, D6, C2xC6, C2xC6, C2xD4, C3xS3, C3:S3, C3:S3, C3xC6, C3xC6, C4xS3, D12, C3:D4, C2xC12, C3xD4, C3xD4, C22xS3, C22xC6, He3, C3xDic3, C3:Dic3, C3xC12, C3xC12, S3xC6, C2xC3:S3, C2xC3:S3, C2xC3:S3, C62, C62, S3xD4, C6xD4, C32:C6, C32:C6, C2xHe3, C2xHe3, S3xC12, C3xD12, C3xC3:D4, C4xC3:S3, C12:S3, C32:7D4, D4xC32, D4xC32, S3xC2xC6, C22xC3:S3, C32:C12, C4xHe3, C2xC32:C6, C2xC32:C6, C2xC32:C6, C22xHe3, C3xS3xD4, D4xC3:S3, C4xC32:C6, He3:4D4, He3:6D4, D4xHe3, C22xC32:C6, D4xC32:C6
Quotients: C1, C2, C3, C22, S3, C6, D4, C23, D6, C2xC6, C2xD4, C3xS3, C3xD4, C22xS3, C22xC6, S3xC6, S3xD4, C6xD4, C32:C6, S3xC2xC6, C2xC32:C6, C3xS3xD4, C22xC32:C6, D4xC32:C6

Smallest permutation representation of D4xC32: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+++++++++++++++
imageC1C2C2C2C2C2C3C6C6C6C6C6D4xC32:C6S3D4D6D6C3xS3C3xD4S3xC6S3xC6S3xD4C3xS3xD4C32:C6C2xC32:C6C2xC32:C6
kernelD4xC32:C6C4xC32:C6He3:4D4He3:6D4D4xHe3C22xC32:C6D4xC3:S3C4xC3:S3C12:S3C32:7D4D4xC32C22xC3:S3C1D4xC32C32:C6C3xC12C62C3xD4C3:S3C12C2xC6C32C3D4C4C22
# reps11121222242411212242412112

Matrix representation of D4xC32: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] >;
 

D4xC32: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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