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G = D4×D12  order 192 = 26·3

Direct product of D4 and D12

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

Aliases: D4×D12, C42⋊15D6, C6.1022+ 1+4, C3⋊1D42, C4⋊4(S3×D4), C4⋊C4⋊46D6, (C3×D4)⋊9D4, D6⋊5(C2×D4), C4⋊1(C2×D12), C12⋊1(C2×D4), (C4×D4)⋊11S3, (C4×D12)⋊27C2, (D4×C12)⋊13C2, D6⋊D4⋊5C2, C12⋊7D4⋊7C2, D6⋊C4⋊4C22, C22⋊C4⋊45D6, C22⋊2(C2×D12), (C22×C4)⋊14D6, C12⋊D4⋊14C2, C4⋊D12⋊11C2, (C4×C12)⋊18C22, (C2×D4).247D6, (C22×D12)⋊8C2, (C2×C6).93C24, C6.15(C22×D4), C2.14(D4○D12), (C2×D12)⋊16C22, (S3×C23)⋊5C22, C4⋊Dic3⋊58C22, (C22×C12)⋊9C22, C2.17(C22×D12), (C2×C12).158C23, (C6×D4).256C22, (C22×C6).163C23, C23.181(C22×S3), C22.118(S3×C23), (C2×Dic3).39C23, (C22×S3).171C23, (C2×S3×D4)⋊3C2, (C2×C6)⋊1(C2×D4), C2.21(C2×S3×D4), (S3×C2×C4)⋊2C22, (C3×C4⋊C4)⋊58C22, (C2×C3⋊D4)⋊2C22, (C3×C22⋊C4)⋊49C22, (C2×C4).157(C22×S3), SmallGroup(192,1108)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C6 — D4×D12
Chief series
C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C23 — C2×S3×D4 — D4×D12
Lower central
C3 — C2×C6 — D4×D12
Upper central
C1 — C22 — C4×D4

Generators and relations for D4×D12
 G = < a,b,c,d | a4=b2=c12=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 1432 in 428 conjugacy classes, 123 normal (29 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C2×C4, C2×C4, C2×C4, D4, D4, C23, C23, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, C4×S3, D12, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C2×C12, C3×D4, C22×S3, C22×S3, C22×C6, C4×D4, C4×D4, C22≀C2, C4⋊D4, C4⋊1D4, C22×D4, C4⋊Dic3, D6⋊C4, C4×C12, C3×C22⋊C4, C3×C4⋊C4, S3×C2×C4, C2×D12, C2×D12, C2×D12, S3×D4, C2×C3⋊D4, C22×C12, C6×D4, S3×C23, D42, C4×D12, C4⋊D12, D6⋊D4, C12⋊D4, C12⋊7D4, D4×C12, C22×D12, C2×S3×D4, D4×D12
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C24, D12, C22×S3, C22×D4, 2+ 1+4, C2×D12, S3×D4, S3×C23, D42, C22×D12, C2×S3×D4, D4○D12, D4×D12

Smallest permutation representation of D4×D12
►On 48 points
Generators in S48
(1 43 16 26)(2 44 17 27)(3 45 18 28)(4 46 19 29)(5 47 20 30)(6 48 21 31)(7 37 22 32)(8 38 23 33)(9 39 24 34)(10 40 13 35)(11 41 14 36)(12 42 15 25)

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

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

45 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J2K2L2M2N2O 3 4A4B4C4D4E4F4G4H4I6A6B6C6D6E6F6G12A12B12C12D12E···12L
order1222222222222222344444444466666661212121212···12
size11112222666612121212222224441212222444422224···4

45 irreducible representations

dim111111111222222222444
type+++++++++++++++++++++
imageC1C2C2C2C2C2C2C2C2S3D4D4D6D6D6D6D6D122+ 1+4S3×D4D4○D12
kernelD4×D12C4×D12C4⋊D12D6⋊D4C12⋊D4C12⋊7D4D4×C12C22×D12C2×S3×D4C4×D4D12C3×D4C42C22⋊C4C4⋊C4C22×C4C2×D4D4C6C4C2
# reps111422122144121218122

Matrix representation of D4×D12 ►in GL6(ℤ)

1-20000
1-10000
001000
000100
0000-10
00000-1
,
1-20000
0-10000
001000
000100
000010
000001
,
100000
010000
001100
00-1000
0000-1-2
000011
,
100000
010000
00-1-100
000100
0000-10
000011

G:=sub<GL(6,Integers())| [1,1,0,0,0,0,-2,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1],[1,0,0,0,0,0,-2,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,-1,0,0,0,0,1,0,0,0,0,0,0,0,-1,1,0,0,0,0,-2,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,-1,1,0,0,0,0,0,0,-1,1,0,0,0,0,0,1] >;
 

D4×D12 in GAP, Magma, Sage, TeX 

D_4\times D_{12}
 
% in TeX
 
G:=Group("D4xD12");
 
// GroupNames label
 
G:=SmallGroup(192,1108);
 
// by ID
 
G=gap.SmallGroup(192,1108);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,387,675,80,6278]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^2=c^12=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
 
// generators/relations
 

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