D4.(C2xD4) - GroupNames

G = D₄.(C₂×D₄) order 128 = 2⁷

8^th non-split extension by D₄ of C₂×D₄ acting via C₂×D₄/C₂³=C₂

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

Aliases: C₄○D₄.₁₃D₄, D₄.₄₄(C₂×D₄), Q₈.₄₄(C₂×D₄), C₄.₄₀C₂²≀C₂, C₂²⋊SD₁₆⋊₃C₂, (C₂×D₄).₁₄₆D₄, C₄⋊C₄.₁₁C₂³, C₂²⋊C₈⋊₈C₂², (C₂×Q₈).₂₃₀D₄, C₄.₄₆(C₂²×D₄), C₂²⋊Q₈⋊₁C₂², D₄.₇D₄⋊₁₅C₂, (C₂×C₄).₂₂₈C₂⁴, (C₂×C₈).₃₀₀C₂³, (C₂²×C₈)⋊₃₅C₂², C₂.₈(D₄○SD₁₆), (C₂×Q₁₆)⋊₁₅C₂², C₂³.₂₃₂(C₂×D₄), (C₂×Q₈).₂₂C₂³, D₄⋊C₄⋊₁₃C₂², C₄²⋊C₂⋊₈C₂², Q₈⋊C₄⋊₇₅C₂², C₂².₂₀C₂²≀C₂, (C₂²×SD₁₆)⋊₁₉C₂, (C₂×SD₁₆)⋊₇₀C₂², (C₂×D₄).₃₈₂C₂³, C₂³.₃₇D₄⋊₅C₂, (C₂×M₄(2))⋊₅C₂², (C₂²×Q₈)⋊₁₄C₂², C₂³.₂₄D₄⋊₂₁C₂, (C₂²×C₄).₂₇₆C₂³, C₂².₄₈₈(C₂²×D₄), (C₂×2₊¹⁺⁴).₇C₂, C₂³.₃₈C₂³⋊₃C₂, (C₂²×D₄).₃₂₆C₂², (C₂×C₄).₄₅₅(C₂×D₄), (C₂×C₈.C₂²)⋊₈C₂, (C₂²×C₈)⋊C₂⋊₄C₂, C₂.₄₆(C₂×C₂²≀C₂), (C₂×C₄○D₄).₁₀₁C₂², SmallGroup(128,1741)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series

C₁ — C₂×C₄ — D₄.(C₂×D₄)

Chief series

C₁ — C₂ — C₂² — C₂×C₄ — C₂²×C₄ — C₂×C₄○D₄ — C₂×2₊¹⁺⁴ — D₄.(C₂×D₄)

Lower central

C₁ — C₂ — C₂×C₄ — D₄.(C₂×D₄)

Upper central

C₁ — C₂² — C₂×C₄○D₄ — D₄.(C₂×D₄)

Jennings

C₁ — C₂ — C₂ — C₂×C₄ — D₄.(C₂×D₄)

Generators and relations for D₄.(C₂×D₄)
G = < a,b,c,d,e | a⁴=b²=1, c²=d⁴=e²=a², bab=eae^-1=a^-1, ac=ca, ad=da, bc=cb, dbd^-1=ebe^-1=a^-1b, cd=dc, ece^-1=a²c, ede^-1=d³ >

Subgroups: 796 in 376 conjugacy classes, 108 normal (28 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₄, C₂², C₂², C₂², C₈, C₂×C₄, C₂×C₄, C₂×C₄, D₄, D₄, Q₈, Q₈, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂×C₈, C₂×C₈, M₄(2), SD₁₆, Q₁₆, C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₂×Q₈, C₂×Q₈, C₄○D₄, C₄○D₄, C₂⁴, C₂²⋊C₈, D₄⋊C₄, Q₈⋊C₄, C₄²⋊C₂, C₂²⋊Q₈, C₂².D₄, C₄.₄D₄, C₄⋊Q₈, C₂²×C₈, C₂×M₄(2), C₂×SD₁₆, C₂×SD₁₆, C₂×Q₁₆, C₈.C₂², C₂²×D₄, C₂²×D₄, C₂²×D₄, C₂²×Q₈, C₂×C₄○D₄, C₂×C₄○D₄, C₂×C₄○D₄, 2₊¹⁺⁴, (C₂²×C₈)⋊C₂, C₂³.₂₄D₄, C₂³.₃₇D₄, C₂²⋊SD₁₆, D₄.₇D₄, C₂³.₃₈C₂³, C₂²×SD₁₆, C₂×C₈.C₂², C₂×2₊¹⁺⁴, D₄.(C₂×D₄)
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₂⁴, C₂²≀C₂, C₂²×D₄, C₂×C₂²≀C₂, D₄○SD₁₆, D₄.(C₂×D₄)

Smallest permutation representation of D₄.(C₂×D₄)
►On 32 points

Generators in S₃₂

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

(2 27)(3 7)(4 25)(6 31)(8 29)(10 20)(11 15)(12 18)(14 24)(16 22)(19 23)(26 30)

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

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

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

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

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

32 conjugacy classes

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

32 irreducible representations

dim

type

image

C₁

C₂

D₄

D₄○SD₁₆

kernel

D₄.(C₂×D₄)

(C₂²×C₈)⋊C₂

C₂³.₂₄D₄

C₂³.₃₇D₄

C₂²⋊SD₁₆

D₄.₇D₄

C₂³.₃₈C₂³

C₂²×SD₁₆

C₂×C₈.C₂²

C₂×2₊¹⁺⁴

C₂×D₄

C₂×Q₈

C₄○D₄

C₂

# reps

Matrix representation of D₄.(C₂×D₄) ►in GL₆(𝔽₁₇)

16	0	0	0	0	0
0	16	0	0	0	0
0	0	16	15	0	0
0	0	1	1	0	0
0	0	0	1	0	1
0	0	16	16	16	0

1	0	0	0	0	0
7	16	0	0	0	0
0	0	1	0	0	0
0	0	16	16	0	0
0	0	16	0	16	0
0	0	0	0	0	1

16	0	0	0	0	0
0	16	0	0	0	0
0	0	1	0	0	15
0	0	16	0	16	1
0	0	0	1	0	1
0	0	1	0	0	16

10	2	0	0	0	0
10	7	0	0	0	0
0	0	10	10	0	0
0	0	12	0	0	0
0	0	0	12	5	12
0	0	5	5	5	5

10	2	0	0	0	0
10	7	0	0	0	0
0	0	10	10	0	0
0	0	12	7	0	0
0	0	0	12	5	12
0	0	5	5	12	12

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,1,0,16,0,0,15,1,1,16,0,0,0,0,0,16,0,0,0,0,1,0],[1,7,0,0,0,0,0,16,0,0,0,0,0,0,1,16,16,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,16,0,1,0,0,0,0,1,0,0,0,0,16,0,0,0,0,15,1,1,16],[10,10,0,0,0,0,2,7,0,0,0,0,0,0,10,12,0,5,0,0,10,0,12,5,0,0,0,0,5,5,0,0,0,0,12,5],[10,10,0,0,0,0,2,7,0,0,0,0,0,0,10,12,0,5,0,0,10,7,12,5,0,0,0,0,5,12,0,0,0,0,12,12] >;

D₄.(C₂×D₄) in GAP, Magma, Sage, TeX

D_4.(C_2\times D_4)

% in TeX

G:=Group("D4.(C2xD4)");

// GroupNames label

G:=SmallGroup(128,1741);

// by ID

G=gap.SmallGroup(128,1741);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,758,521,2804,1411,172]);

// Polycyclic

G:=Group<a,b,c,d,e|a^4=b^2=1,c^2=d^4=e^2=a^2,b*a*b=e*a*e^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=e*b*e^-1=a^-1*b,c*d=d*c,e*c*e^-1=a^2*c,e*d*e^-1=d^3>;

// generators/relations

G = D4.(C2×D4) order 128 = 27

8th non-split extension by D4 of C2×D4 acting via C2×D4/C23=C2

G = D₄.(C₂×D₄) order 128 = 2⁷

8^th non-split extension by D₄ of C₂×D₄ acting via C₂×D₄/C₂³=C₂