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## G = D4×D12order 192 = 26·3

### Direct product of D4 and D12

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, C41D4, 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, C127D4, 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 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 2L 2M 2N 2O 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 6A 6B 6C 6D 6E 6F 6G 12A 12B 12C 12D 12E ··· 12L order 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 4 4 4 4 4 4 4 4 4 6 6 6 6 6 6 6 12 12 12 12 12 ··· 12 size 1 1 1 1 2 2 2 2 6 6 6 6 12 12 12 12 2 2 2 2 2 4 4 4 12 12 2 2 2 4 4 4 4 2 2 2 2 4 ··· 4

45 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 S3 D4 D4 D6 D6 D6 D6 D6 D12 2+ 1+4 S3×D4 D4○D12 kernel D4×D12 C4×D12 C4⋊D12 D6⋊D4 C12⋊D4 C12⋊7D4 D4×C12 C22×D12 C2×S3×D4 C4×D4 D12 C3×D4 C42 C22⋊C4 C4⋊C4 C22×C4 C2×D4 D4 C6 C4 C2 # reps 1 1 1 4 2 2 1 2 2 1 4 4 1 2 1 2 1 8 1 2 2

Matrix representation of D4×D12 in GL6(ℤ)

 1 -2 0 0 0 0 1 -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 -2 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 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 -2 0 0 0 0 1 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 -1 -1 0 0 0 0 0 1 0 0 0 0 0 0 -1 0 0 0 0 0 1 1

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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