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G = D4×C3⋊D4  order 192 = 26·3

Direct product of D4 and C3⋊D4

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

Aliases: D4×C3⋊D4, C24⋊11D6, C6.882+ 1+4, C3⋊5D42, D6⋊9(C2×D4), C12⋊8(C2×D4), (C3×D4)⋊16D4, (C2×D4)⋊37D6, C22⋊5(S3×D4), Dic3⋊5(C2×D4), (C22×C4)⋊30D6, D6⋊3D4⋊39C2, C23⋊2D6⋊29C2, C12⋊3D4⋊28C2, C12⋊7D4⋊37C2, D6⋊C4⋊35C22, (D4×Dic3)⋊38C2, (C22×D4)⋊10S3, (C6×D4)⋊56C22, C24⋊4S3⋊11C2, (C2×D12)⋊38C22, (C2×C6).296C24, C4⋊Dic3⋊44C22, (C23×C6)⋊13C22, C6.143(C22×D4), C23.14D6⋊40C2, C2.91(D4⋊6D6), (C2×C12).543C23, Dic3⋊C4⋊73C22, (S3×C23)⋊14C22, (C22×C12)⋊23C22, (C4×Dic3)⋊41C22, C6.D4⋊62C22, C22.309(S3×C23), C23.215(C22×S3), (C22×C6).230C23, (C22×S3).240C23, (C2×Dic3).153C23, (C22×Dic3)⋊33C22, (D4×C2×C6)⋊5C2, (C2×S3×D4)⋊25C2, (C2×C6)⋊8(C2×D4), C4⋊2(C2×C3⋊D4), C2.103(C2×S3×D4), (C4×C3⋊D4)⋊25C2, (S3×C2×C4)⋊30C22, C22⋊3(C2×C3⋊D4), (C22×C3⋊D4)⋊14C2, (C2×C3⋊D4)⋊45C22, C2.16(C22×C3⋊D4), (C2×C4).626(C22×S3), SmallGroup(192,1360)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C6 — D4×C3⋊D4
Chief series
C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C23 — C2×S3×D4 — D4×C3⋊D4
Lower central
C3 — C2×C6 — D4×C3⋊D4
Upper central
C1 — C22 — C22×D4

Generators and relations for D4×C3⋊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, C2×C4, C2×C4, D4, D4, C23, C23, C23, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C24, C24, C4×S3, D12, C2×Dic3, C2×Dic3, C2×Dic3, C3⋊D4, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×D4, C22×S3, C22×S3, C22×S3, C22×C6, C22×C6, C22×C6, C4×D4, C22≀C2, C4⋊D4, C4⋊1D4, C22×D4, C22×D4, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, D6⋊C4, C6.D4, C6.D4, S3×C2×C4, C2×D12, S3×D4, C22×Dic3, C2×C3⋊D4, C2×C3⋊D4, C2×C3⋊D4, C22×C12, C6×D4, C6×D4, C6×D4, S3×C23, C23×C6, D42, C4×C3⋊D4, C12⋊7D4, D4×Dic3, C23⋊2D6, D6⋊3D4, C23.14D6, C12⋊3D4, C24⋊4S3, C2×S3×D4, C22×C3⋊D4, D4×C2×C6, D4×C3⋊D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C24, C3⋊D4, C22×S3, C22×D4, 2+ 1+4, S3×D4, C2×C3⋊D4, S3×C23, D42, C2×S3×D4, D4⋊6D6, C22×C3⋊D4, D4×C3⋊D4

Smallest permutation representation of D4×C3⋊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+4S3×D4D4⋊6D6
kernelD4×C3⋊D4C4×C3⋊D4C12⋊7D4D4×Dic3C23⋊2D6D6⋊3D4C23.14D6C12⋊3D4C24⋊4S3C2×S3×D4C22×C3⋊D4D4×C2×C6C22×D4C3⋊D4C3×D4C22×C4C2×D4C24D4C6C22C2
# reps1111212121211441428122

Matrix representation of D4×C3⋊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] >;
 

D4×C3⋊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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