C2xD4.4D4 - GroupNames

G = C₂×D₄.₄D₄ order 128 = 2⁷

Direct product of C₂ and D₄.₄D₄

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

Aliases: C₂×D₄.₄D₄, M₄(2).₈C₂³, C₄○D₄.₃₂D₄, D₄.₁₂(C₂×D₄), (C₂×C₈).₃₅₁D₄, C₈.₁₁₂(C₂×D₄), Q₈.₁₂(C₂×D₄), (C₂×D₄).₂₂₄D₄, C₈○D₄⋊₁₂C₂², (C₂×D₈)⋊₄₈C₂², (C₂²×D₈)⋊₁₇C₂, (C₂×C₄).₁₆C₂⁴, (C₂×Q₈).₁₇₉D₄, C₈⋊C₂²⋊₁₀C₂², (C₂×C₈).₂₆₁C₂³, C₄○D₄.₂₈C₂³, (C₂×D₄).₇₁C₂³, C₄.₁₆₃(C₂²×D₄), C₄.₁₁₄(C₄⋊D₄), C₈.C₄⋊₁₆C₂², C₄.D₄⋊₁₃C₂², C₂³.₃₁₇(C₄○D₄), C₂².₉₀(C₄⋊D₄), (C₂²×C₄).₉₉₁C₂³, (C₂²×C₈).₂₆₅C₂², (C₂²×D₄).₃₅₃C₂², (C₂×M₄(2)).₅₉C₂², (C₂×C₈○D₄)⋊₈C₂, (C₂×C₈⋊C₂²)⋊₂₀C₂, C₂.₈₇(C₂×C₄⋊D₄), (C₂×C₈.C₄)⋊₂₆C₂, (C₂×C₄).₁₄₂₈(C₂×D₄), (C₂×C₄.D₄)⋊₁₁C₂, C₂².₁₉(C₂×C₄○D₄), (C₂×C₄).₂₉₀(C₄○D₄), (C₂×C₄○D₄).₃₀₅C₂², SmallGroup(128,1797)

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

Derived series

C₁ — C₂×C₄ — C₂×D₄.₄D₄

Chief series

C₁ — C₂ — C₄ — C₂×C₄ — C₂²×C₄ — C₂×C₄○D₄ — C₂×C₈○D₄ — C₂×D₄.₄D₄

Lower central

C₁ — C₂ — C₂×C₄ — C₂×D₄.₄D₄

Upper central

C₁ — C₂² — C₂²×C₄ — C₂×D₄.₄D₄

Jennings

C₁ — C₂ — C₂ — C₂×C₄ — C₂×D₄.₄D₄

Generators and relations for C₂×D₄.₄D₄
G = < a,b,c,d,e | a²=b⁴=c²=e²=1, d⁴=b², ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe=b^-1, bd=db, cd=dc, ece=bc, ede=b²d³ >

Subgroups: 540 in 244 conjugacy classes, 100 normal (28 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₂², C₂², C₈, C₈, C₂×C₄, C₂×C₄, D₄, D₄, Q₈, Q₈, C₂³, C₂³, C₂×C₈, C₂×C₈, C₂×C₈, M₄(2), M₄(2), D₈, SD₁₆, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₄○D₄, C₄○D₄, C₂⁴, C₄.D₄, C₈.C₄, C₂²×C₈, C₂²×C₈, C₂×M₄(2), C₂×M₄(2), C₂×M₄(2), C₈○D₄, C₈○D₄, C₂×D₈, C₂×D₈, C₂×SD₁₆, C₈⋊C₂², C₈⋊C₂², C₂²×D₄, C₂×C₄○D₄, C₂×C₄.D₄, C₂×C₈.C₄, D₄.₄D₄, C₂×C₈○D₄, C₂²×D₈, C₂×C₈⋊C₂², C₂×D₄.₄D₄
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₄⋊D₄, C₂²×D₄, C₂×C₄○D₄, D₄.₄D₄, C₂×C₄⋊D₄, C₂×D₄.₄D₄

Smallest permutation representation of C₂×D₄.₄D₄
►On 32 points

Generators in S₃₂

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

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

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

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

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

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

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

32 conjugacy classes

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

32 irreducible representations

dim

type

image

C₁

C₂

D₄

C₄○D₄

D₄.₄D₄

kernel

C₂×D₄.₄D₄

C₂×C₄.D₄

C₂×C₈.C₄

D₄.₄D₄

C₂×C₈○D₄

C₂²×D₈

C₂×C₈⋊C₂²

C₂×C₈

C₂×D₄

C₂×Q₈

C₄○D₄

C₂×C₄

C₂³

C₂

# reps

Matrix representation of C₂×D₄.₄D₄ ►in GL₆(𝔽₁₇)

16	0	0	0	0	0
0	16	0	0	0	0
0	0	16	0	0	0
0	0	0	16	0	0
0	0	0	0	16	0
0	0	0	0	0	16

16	0	0	0	0	0
0	16	0	0	0	0
0	0	0	1	0	0
0	0	16	0	0	0
0	0	1	1	1	2
0	0	0	16	16	16

1	15	0	0	0	0
0	16	0	0	0	0
0	0	1	1	1	2
0	0	0	0	1	0
0	0	0	1	0	0
0	0	0	16	16	16

13	8	0	0	0	0
0	4	0	0	0	0
0	0	14	14	0	0
0	0	3	14	0	0
0	0	3	3	0	6
0	0	14	0	14	11

16	0	0	0	0	0
16	1	0	0	0	0
0	0	0	16	0	0
0	0	16	0	0	0
0	0	0	0	1	0
0	0	0	0	16	16

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,16,1,0,0,0,1,0,1,16,0,0,0,0,1,16,0,0,0,0,2,16],[1,0,0,0,0,0,15,16,0,0,0,0,0,0,1,0,0,0,0,0,1,0,1,16,0,0,1,1,0,16,0,0,2,0,0,16],[13,0,0,0,0,0,8,4,0,0,0,0,0,0,14,3,3,14,0,0,14,14,3,0,0,0,0,0,0,14,0,0,0,0,6,11],[16,16,0,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,16,0,0,0,0,0,0,0,1,16,0,0,0,0,0,16] >;

C₂×D₄.₄D₄ in GAP, Magma, Sage, TeX

C_2\times D_4._4D_4

% in TeX

G:=Group("C2xD4.4D4");

// GroupNames label

G:=SmallGroup(128,1797);

// by ID

G=gap.SmallGroup(128,1797);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,568,758,2804,172,4037,1027,124]);

// Polycyclic

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

// generators/relations

G = C2×D4.4D4 order 128 = 27

Direct product of C2 and D4.4D4

G = C₂×D₄.₄D₄ order 128 = 2⁷

Direct product of C₂ and D₄.₄D₄