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G = D4xC3:D4  order 192 = 26·3

Direct product of D4 and C3:D4

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D4xC3:D4, C24:11D6, C6.882+ 1+4, C3:5D42, D6:9(C2xD4), C12:8(C2xD4), (C3xD4):16D4, (C2xD4):37D6, C22:5(S3xD4), Dic3:5(C2xD4), (C22xC4):30D6, D6:3D4:39C2, C23:2D6:29C2, C12:3D4:28C2, C12:7D4:37C2, D6:C4:35C22, (D4xDic3):38C2, (C22xD4):10S3, (C6xD4):56C22, C24:4S3:11C2, (C2xD12):38C22, (C2xC6).296C24, C4:Dic3:44C22, (C23xC6):13C22, C6.143(C22xD4), C23.14D6:40C2, C2.91(D4:6D6), (C2xC12).543C23, Dic3:C4:73C22, (S3xC23):14C22, (C22xC12):23C22, (C4xDic3):41C22, C6.D4:62C22, C22.309(S3xC23), C23.215(C22xS3), (C22xC6).230C23, (C22xS3).240C23, (C2xDic3).153C23, (C22xDic3):33C22, (D4xC2xC6):5C2, (C2xS3xD4):25C2, (C2xC6):8(C2xD4), C4:2(C2xC3:D4), C2.103(C2xS3xD4), (C4xC3:D4):25C2, (S3xC2xC4):30C22, C22:3(C2xC3:D4), (C22xC3:D4):14C2, (C2xC3:D4):45C22, C2.16(C22xC3:D4), (C2xC4).626(C22xS3), SmallGroup(192,1360)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2xC6 — D4xC3:D4
Chief series
C1 — C3 — C6 — C2xC6 — C22xS3 — S3xC23 — C2xS3xD4 — D4xC3:D4
Lower central
C3 — C2xC6 — D4xC3:D4
Upper central
C1 — C22 — C22xD4

Generators and relations for D4xC3:D4
 G = < a,b,c,d,e | a4=b2=c3=d4=e2=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=ece=c-1, ede=d-1 >

Subgroups: 1240 in 428 conjugacy classes, 123 normal (43 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C2xC4, C2xC4, D4, D4, C23, C23, C23, Dic3, Dic3, C12, C12, D6, D6, C2xC6, C2xC6, C2xC6, C42, C22:C4, C4:C4, C22xC4, C22xC4, C2xD4, C2xD4, C2xD4, C24, C24, C4xS3, D12, C2xDic3, C2xDic3, C2xDic3, C3:D4, C3:D4, C2xC12, C2xC12, C3xD4, C3xD4, C22xS3, C22xS3, C22xS3, C22xC6, C22xC6, C22xC6, C4xD4, C22wrC2, C4:D4, C4:1D4, C22xD4, C22xD4, C4xDic3, Dic3:C4, C4:Dic3, D6:C4, D6:C4, C6.D4, C6.D4, S3xC2xC4, C2xD12, S3xD4, C22xDic3, C2xC3:D4, C2xC3:D4, C2xC3:D4, C22xC12, C6xD4, C6xD4, C6xD4, S3xC23, C23xC6, D42, C4xC3:D4, C12:7D4, D4xDic3, C23:2D6, D6:3D4, C23.14D6, C12:3D4, C24:4S3, C2xS3xD4, C22xC3:D4, D4xC2xC6, D4xC3:D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2xD4, C24, C3:D4, C22xS3, C22xD4, 2+ 1+4, S3xD4, C2xC3:D4, S3xC23, D42, C2xS3xD4, D4:6D6, C22xC3:D4, D4xC3:D4

Smallest permutation representation of D4xC3:D4
►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)

(2 4)(5 7)(9 11)(13 15)(18 20)(21 23)(25 27)(29 31)(34 36)(37 39)(42 44)(46 48)

(1 17 16)(2 18 13)(3 19 14)(4 20 15)(5 34 29)(6 35 30)(7 36 31)(8 33 32)(9 25 44)(10 26 41)(11 27 42)(12 28 43)(21 37 48)(22 38 45)(23 39 46)(24 40 47)

(1 6 41 47)(2 7 42 48)(3 8 43 45)(4 5 44 46)(9 39 20 29)(10 40 17 30)(11 37 18 31)(12 38 19 32)(13 36 27 21)(14 33 28 22)(15 34 25 23)(16 35 26 24)

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

45 conjugacy classes

class 1 2A2B2C2D···2I2J2K2L2M2N2O 3 4A4B4C4D4E4F4G4H4I6A···6G6H···6O12A12B12C12D
order12222···222222234444444446···66···612121212
size11112···244661212222466121212122···24···44444

45 irreducible representations

dim1111111111112222222444
type++++++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2S3D4D4D6D6D6C3:D42+ 1+4S3xD4D4:6D6
kernelD4xC3:D4C4xC3:D4C12:7D4D4xDic3C23:2D6D6:3D4C23.14D6C12:3D4C24:4S3C2xS3xD4C22xC3:D4D4xC2xC6C22xD4C3:D4C3xD4C22xC4C2xD4C24D4C6C22C2
# reps1111212121211441428122

Matrix representation of D4xC3:D4 ►in GL4(F13) generated by

12000
01200
00012
0010
,
1000
0100
0010
00012
,
121200
1000
0010
0001
,
2400
21100
0010
0001
,
1000
121200
00120
00012
G:=sub<GL(4,GF(13))| [12,0,0,0,0,12,0,0,0,0,0,1,0,0,12,0],[1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,12],[12,1,0,0,12,0,0,0,0,0,1,0,0,0,0,1],[2,2,0,0,4,11,0,0,0,0,1,0,0,0,0,1],[1,12,0,0,0,12,0,0,0,0,12,0,0,0,0,12] >;
 

D4xC3:D4 in GAP, Magma, Sage, TeX 

D_4\times C_3\rtimes D_4
 
% in TeX
 
G:=Group("D4xC3:D4");
 
// GroupNames label
 
G:=SmallGroup(192,1360);
 
// by ID
 
G=gap.SmallGroup(192,1360);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,387,675,6278]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^4=b^2=c^3=d^4=e^2=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,d*c*d^-1=e*c*e=c^-1,e*d*e=d^-1>;
 
// generators/relations
 

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