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## G = D4×D8order 128 = 27

### Direct product of D4 and D8

direct product, p-group, metabelian, nilpotent (class 3), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — D4×D8
 Chief series C1 — C2 — C22 — C2×C4 — C2×D4 — C22×D4 — C22×D8 — D4×D8
 Lower central C1 — C2 — C2×C4 — D4×D8
 Upper central C1 — C22 — C4×D4 — D4×D8
 Jennings C1 — C2 — C2 — C2×C4 — D4×D8

Generators and relations for D4×D8
G = < a,b,c,d | a4=b2=c8=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 824 in 310 conjugacy classes, 104 normal (24 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, D4, D4, C23, C23, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C2×C8, D8, D8, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C24, C4×C8, C22⋊C8, D4⋊C4, C4⋊C8, C2.D8, C4×D4, C4×D4, C22≀C2, C4⋊D4, C4⋊D4, C41D4, C22×C8, C2×D8, C2×D8, C2×D8, C22×D4, C22×D4, C8×D4, C4×D8, C22⋊D8, C4⋊D8, C87D4, C84D4, D42, C22×D8, D4×D8
Quotients: C1, C2, C22, D4, C23, D8, C2×D4, C24, C2×D8, C22×D4, 2+ 1+4, D42, C22×D8, D4○D8, D4×D8

Smallest permutation representation of D4×D8
On 32 points
Generators in S32
(1 11 24 32)(2 12 17 25)(3 13 18 26)(4 14 19 27)(5 15 20 28)(6 16 21 29)(7 9 22 30)(8 10 23 31)
(1 28)(2 29)(3 30)(4 31)(5 32)(6 25)(7 26)(8 27)(9 18)(10 19)(11 20)(12 21)(13 22)(14 23)(15 24)(16 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)
(1 23)(2 22)(3 21)(4 20)(5 19)(6 18)(7 17)(8 24)(9 25)(10 32)(11 31)(12 30)(13 29)(14 28)(15 27)(16 26)

G:=sub<Sym(32)| (1,11,24,32)(2,12,17,25)(3,13,18,26)(4,14,19,27)(5,15,20,28)(6,16,21,29)(7,9,22,30)(8,10,23,31), (1,28)(2,29)(3,30)(4,31)(5,32)(6,25)(7,26)(8,27)(9,18)(10,19)(11,20)(12,21)(13,22)(14,23)(15,24)(16,17), (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), (1,23)(2,22)(3,21)(4,20)(5,19)(6,18)(7,17)(8,24)(9,25)(10,32)(11,31)(12,30)(13,29)(14,28)(15,27)(16,26)>;

G:=Group( (1,11,24,32)(2,12,17,25)(3,13,18,26)(4,14,19,27)(5,15,20,28)(6,16,21,29)(7,9,22,30)(8,10,23,31), (1,28)(2,29)(3,30)(4,31)(5,32)(6,25)(7,26)(8,27)(9,18)(10,19)(11,20)(12,21)(13,22)(14,23)(15,24)(16,17), (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), (1,23)(2,22)(3,21)(4,20)(5,19)(6,18)(7,17)(8,24)(9,25)(10,32)(11,31)(12,30)(13,29)(14,28)(15,27)(16,26) );

G=PermutationGroup([[(1,11,24,32),(2,12,17,25),(3,13,18,26),(4,14,19,27),(5,15,20,28),(6,16,21,29),(7,9,22,30),(8,10,23,31)], [(1,28),(2,29),(3,30),(4,31),(5,32),(6,25),(7,26),(8,27),(9,18),(10,19),(11,20),(12,21),(13,22),(14,23),(15,24),(16,17)], [(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)], [(1,23),(2,22),(3,21),(4,20),(5,19),(6,18),(7,17),(8,24),(9,25),(10,32),(11,31),(12,30),(13,29),(14,28),(15,27),(16,26)]])

35 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 2L 2M 2N 2O 4A 4B 4C 4D 4E 4F 4G 4H 4I 8A 8B 8C 8D 8E ··· 8J order 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 8 8 8 8 8 ··· 8 size 1 1 1 1 2 2 2 2 4 4 4 4 8 8 8 8 2 2 2 2 4 4 4 8 8 2 2 2 2 4 ··· 4

35 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 4 4 type + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 D4 D4 D4 D4 D8 2+ 1+4 D4○D8 kernel D4×D8 C8×D4 C4×D8 C22⋊D8 C4⋊D8 C8⋊7D4 C8⋊4D4 D42 C22×D8 C22⋊C4 C4⋊C4 D8 C2×D4 D4 C4 C2 # reps 1 1 1 4 2 2 1 2 2 2 1 4 1 8 1 2

Matrix representation of D4×D8 in GL4(𝔽17) generated by

 16 0 0 0 0 16 0 0 0 0 14 2 0 0 12 3
,
 1 0 0 0 0 1 0 0 0 0 14 2 0 0 13 3
,
 14 3 0 0 14 14 0 0 0 0 16 0 0 0 0 16
,
 14 3 0 0 3 3 0 0 0 0 1 0 0 0 0 1
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,14,12,0,0,2,3],[1,0,0,0,0,1,0,0,0,0,14,13,0,0,2,3],[14,14,0,0,3,14,0,0,0,0,16,0,0,0,0,16],[14,3,0,0,3,3,0,0,0,0,1,0,0,0,0,1] >;

D4×D8 in GAP, Magma, Sage, TeX

D_4\times D_8
% in TeX

G:=Group("D4xD8");
// GroupNames label

G:=SmallGroup(128,2011);
// by ID

G=gap.SmallGroup(128,2011);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,2019,346,2804,1411,375,172]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^2=c^8=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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