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## G = D4×C22⋊C4order 128 = 27

### Direct product of D4 and C22⋊C4

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — D4×C22⋊C4
 Chief series C1 — C2 — C22 — C23 — C24 — C25 — D4×C23 — D4×C22⋊C4
 Lower central C1 — C22 — D4×C22⋊C4
 Upper central C1 — C23 — D4×C22⋊C4
 Jennings C1 — C23 — D4×C22⋊C4

Generators and relations for D4×C22⋊C4
G = < a,b,c,d,e | a4=b2=c2=d2=e4=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ece-1=cd=dc, de=ed >

Subgroups: 1292 in 626 conjugacy classes, 196 normal (20 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, D4, D4, C23, C23, C23, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, C24, C24, C2.C42, C2×C42, C2×C22⋊C4, C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C4×D4, C23×C4, C23×C4, C22×D4, C22×D4, C22×D4, C25, C4×C22⋊C4, C243C4, C23.7Q8, C23.23D4, C24.3C22, C22×C22⋊C4, C2×C4×D4, D4×C23, D4×C22⋊C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C4○D4, C24, C2×C22⋊C4, C4×D4, C23×C4, C22×D4, C2×C4○D4, 2+ 1+4, C22×C22⋊C4, C2×C4×D4, C22.11C24, D42, D45D4, D4×C22⋊C4

Smallest permutation representation of D4×C22⋊C4
On 32 points
Generators in S32
(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 4)(2 3)(5 6)(7 8)(9 10)(11 12)(13 16)(14 15)(17 18)(19 20)(21 24)(22 23)(25 28)(26 27)(29 30)(31 32)
(5 14)(6 15)(7 16)(8 13)(9 31)(10 32)(11 29)(12 30)
(1 23)(2 24)(3 21)(4 22)(5 14)(6 15)(7 16)(8 13)(9 31)(10 32)(11 29)(12 30)(17 28)(18 25)(19 26)(20 27)
(1 8 27 10)(2 5 28 11)(3 6 25 12)(4 7 26 9)(13 20 32 23)(14 17 29 24)(15 18 30 21)(16 19 31 22)

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

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

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

50 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2S 2T 2U 2V 2W 4A ··· 4L 4M ··· 4Z order 1 2 ··· 2 2 ··· 2 2 2 2 2 4 ··· 4 4 ··· 4 size 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 2 2 2 4 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C4 D4 D4 C4○D4 2+ 1+4 kernel D4×C22⋊C4 C4×C22⋊C4 C24⋊3C4 C23.7Q8 C23.23D4 C24.3C22 C22×C22⋊C4 C2×C4×D4 D4×C23 C22×D4 C22⋊C4 C2×D4 C23 C22 # reps 1 1 2 1 4 2 2 2 1 16 4 8 4 2

Matrix representation of D4×C22⋊C4 in GL6(ℤ)

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

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

D4×C22⋊C4 in GAP, Magma, Sage, TeX

D_4\times C_2^2\rtimes C_4
% in TeX

G:=Group("D4xC2^2:C4");
// GroupNames label

G:=SmallGroup(128,1070);
// by ID

G=gap.SmallGroup(128,1070);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,758,346]);
// Polycyclic

G:=Group<a,b,c,d,e|a^4=b^2=c^2=d^2=e^4=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,e*c*e^-1=c*d=d*c,d*e=e*d>;
// generators/relations

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