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## G = D4×M4(2)  order 128 = 27

### Direct product of D4 and M4(2)

direct product, p-group, metabelian, nilpotent (class 2), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — D4×M4(2)
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C2×M4(2) — C22×M4(2) — D4×M4(2)
 Lower central C1 — C22 — D4×M4(2)
 Upper central C1 — C2×C4 — D4×M4(2)
 Jennings C1 — C2 — C2 — C2×C4 — D4×M4(2)

Generators and relations for D4×M4(2)
G = < a,b,c,d | a4=b2=c8=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c5 >

Subgroups: 420 in 272 conjugacy classes, 150 normal (38 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, D4, D4, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C24, C4×C8, C8⋊C4, C22⋊C8, C4⋊C8, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C4×D4, C4×D4, C22×C8, C2×M4(2), C2×M4(2), C2×M4(2), C23×C4, C22×D4, C4×M4(2), C24.4C4, C4⋊M4(2), C8×D4, C89D4, C86D4, C2×C4×D4, C22×M4(2), D4×M4(2)
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, M4(2), C22×C4, C2×D4, C4○D4, C24, C4×D4, C2×M4(2), C23×C4, C22×D4, C2×C4○D4, C2×C4×D4, C22×M4(2), Q8○M4(2), D4×M4(2)

Smallest permutation representation of D4×M4(2)
On 32 points
Generators in S32
(1 10 23 26)(2 11 24 27)(3 12 17 28)(4 13 18 29)(5 14 19 30)(6 15 20 31)(7 16 21 32)(8 9 22 25)
(1 5)(2 6)(3 7)(4 8)(9 29)(10 30)(11 31)(12 32)(13 25)(14 26)(15 27)(16 28)(17 21)(18 22)(19 23)(20 24)
(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 20)(3 17)(4 22)(5 19)(6 24)(7 21)(8 18)(9 29)(10 26)(11 31)(12 28)(13 25)(14 30)(15 27)(16 32)

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

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

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

50 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2I 2J 2K 4A 4B 4C 4D 4E ··· 4N 4O 4P 4Q 4R 8A ··· 8H 8I ··· 8T order 1 2 2 2 2 ··· 2 2 2 4 4 4 4 4 ··· 4 4 4 4 4 8 ··· 8 8 ··· 8 size 1 1 1 1 2 ··· 2 4 4 1 1 1 1 2 ··· 2 4 4 4 4 2 ··· 2 4 ··· 4

50 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 4 type + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C4 C4 C4 C4 D4 C4○D4 M4(2) Q8○M4(2) kernel D4×M4(2) C4×M4(2) C24.4C4 C4⋊M4(2) C8×D4 C8⋊9D4 C8⋊6D4 C2×C4×D4 C22×M4(2) C2×C22⋊C4 C2×C4⋊C4 C4×D4 C22×D4 M4(2) C2×C4 D4 C2 # reps 1 1 2 1 2 4 2 1 2 4 2 8 2 4 4 8 2

Matrix representation of D4×M4(2) in GL4(𝔽17) generated by

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

D4×M4(2) in GAP, Magma, Sage, TeX

D_4\times M_4(2)
% in TeX

G:=Group("D4xM4(2)");
// GroupNames label

G:=SmallGroup(128,1666);
// by ID

G=gap.SmallGroup(128,1666);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,184,2019,521,124]);
// 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^5>;
// generators/relations

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