D4xC2^2:C4 - GroupNames

G = D₄×C₂²⋊C₄ order 128 = 2⁷

Direct product of D₄ and C₂²⋊C₄

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

Aliases: D₄×C₂²⋊C₄, C₂⁵.₂₂C₂², C₂³.₂₂₀C₂⁴, C₂⁴.₂₀₆C₂³, C₂².₅₈2₊¹⁺⁴, C₂.₁D₄², C₂²⋊₅(C₄×D₄), C₂⁴⋊₁₁(C₂×C₄), C₂⁴⋊₃C₄⋊₇C₂, (C₂×D₄).₃₃₉D₄, (C₂²×D₄)⋊₂₃C₄, (C₂³×C₄)⋊₆C₂², (D₄×C₂³).₉C₂, C₂.₁(D₄⋊₅D₄), (C₂×C₄²)⋊₁₅C₂², C₂³.₄₁₁(C₂×D₄), C₂³.₇Q₈⋊₁₉C₂, C₂².₉₉(C₂²×D₄), C₂³.₂₈₆(C₄○D₄), C₂³.₂₃D₄⋊₁₀C₂, C₂³.₂₁₃(C₂²×C₄), C₂².₁₁₁(C₂³×C₄), (C₂²×C₄).₁₂₄₁C₂³, C₂⁴.₃C₂²⋊₁₇C₂, C₂.C₄²⋊₁₁C₂², (C₂²×D₄).₆₀₇C₂², C₂.₂₃(C₂².₁₁C₂⁴), (C₂×C₄×D₄)⋊₉C₂, C₂.₂₈(C₂×C₄×D₄), (C₂×C₄)⋊₁₇(C₂×D₄), C₄⋊₁(C₂×C₂²⋊C₄), (C₂×D₄)⋊₄₁(C₂×C₄), (C₄×C₂²⋊C₄)⋊₃₆C₂, (C₂²×C₄)⋊₂₈(C₂×C₄), C₂²⋊₁(C₂×C₂²⋊C₄), (C₂×C₄⋊C₄)⋊₁₀₃C₂², (C₂²×C₂²⋊C₄)⋊₈C₂, (C₂×C₂²⋊C₄)⋊₇₃C₂², (C₂×C₄).₄₅₂(C₂²×C₄), C₂².₁₀₅(C₂×C₄○D₄), C₂.₁₆(C₂²×C₂²⋊C₄), (C₂×D₄)○(C₂×C₂²⋊C₄), SmallGroup(128,1070)

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, C₂⁴⋊₃C₄, 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, D₄⋊₅D₄, 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

type

image

C₁

C₂

C₄

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

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

G = D4×C22⋊C4 order 128 = 27

Direct product of D4 and C22⋊C4

G = D₄×C₂²⋊C₄ order 128 = 2⁷

Direct product of D₄ and C₂²⋊C₄