C2xD4.8D4 - GroupNames

G = C₂xD₄.₈D₄ order 128 = 2⁷

Direct product of C₂ and D₄.₈D₄

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

Aliases: C₂xD₄.₈D₄, C₄².₃₁₁C₂³, M₄(2).₁C₂³, 2_-¹⁺⁴:₅C₂², C₄wrC₂:₉C₂², C₄oD₄.₁₆D₄, D₄.₅₀(C₂xD₄), (C₂xC₄).₈C₂⁴, Q₈.₅₀(C₂xD₄), C₄.₄₅C₂²wrC₂, (C₂xD₄).₂₉₈D₄, C₈:C₂²:₅C₂², (C₂xQ₈).₂₃₃D₄, C₄oD₄.₃C₂³, C₄.₅₃(C₂²xD₄), (C₂xD₄).₃₄C₂³, C₂³.₆₅₀(C₂xD₄), (C₂²xC₄).₁₀₈D₄, (C₂xQ₈).₂₅C₂³, C₄.₁₀D₄:₆C₂², C₄.₄D₄:₅₀C₂², (C₂x2_-¹⁺⁴):₃C₂, C₂².₃₂(C₂²xD₄), C₂².₁₂₀C₂²wrC₂, (C₂²xC₄).₉₆₇C₂³, (C₂xC₄²).₈₀₀C₂², (C₂²xD₄).₃₃₀C₂², (C₂xM₄(2)).₄₄C₂², (C₂²xQ₈).₂₆₄C₂², (C₂xC₄wrC₂):₆C₂, (C₂xC₄).₂₃(C₂xD₄), (C₂xC₈:C₂²):₁₂C₂, C₂.₅₃(C₂xC₂²wrC₂), (C₂xC₄.₄D₄):₃₇C₂, (C₂xC₄.₁₀D₄):₇C₂, (C₂xC₄oD₄).₁₀₆C₂², SmallGroup(128,1748)

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

Derived series

C₁ — C₂xC₄ — C₂xD₄.₈D₄

Chief series

C₁ — C₂ — C₂² — C₂xC₄ — C₂²xC₄ — C₂²xQ₈ — C₂x2_-¹⁺⁴ — C₂xD₄.₈D₄

Lower central

C₁ — C₂ — C₂xC₄ — C₂xD₄.₈D₄

Upper central

C₁ — C₂² — C₂²xC₄ — C₂xD₄.₈D₄

Jennings

C₁ — C₂ — C₂ — C₂xC₄ — C₂xD₄.₈D₄

Generators and relations for C₂xD₄.₈D₄
G = < a,b,c,d,e | a²=b⁴=c²=e²=1, d⁴=b², ab=ba, ac=ca, ad=da, ae=ea, cbc=dbd^-1=ebe=b^-1, dcd^-1=bc, ece=b^-1c, ede=b²d³ >

Subgroups: 708 in 364 conjugacy classes, 108 normal (16 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₂², C₂², C₈, C₂xC₄, C₂xC₄, C₂xC₄, D₄, D₄, Q₈, Q₈, C₂³, C₂³, C₄², C₄², C₂²:C₄, C₂xC₈, M₄(2), M₄(2), D₈, SD₁₆, C₂²xC₄, C₂²xC₄, C₂²xC₄, C₂xD₄, C₂xD₄, C₂xQ₈, C₂xQ₈, C₄oD₄, C₄oD₄, C₂⁴, C₄.₁₀D₄, C₄wrC₂, C₂xC₄², C₂xC₂²:C₄, C₄.₄D₄, C₄.₄D₄, C₂xM₄(2), C₂xD₈, C₂xSD₁₆, C₈:C₂², C₈:C₂², C₂²xD₄, C₂²xQ₈, C₂²xQ₈, C₂xC₄oD₄, C₂xC₄oD₄, 2_-¹⁺⁴, 2_-¹⁺⁴, C₂xC₄.₁₀D₄, C₂xC₄wrC₂, D₄.₈D₄, C₂xC₄.₄D₄, C₂xC₈:C₂², C₂x2_-¹⁺⁴, C₂xD₄.₈D₄
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂xD₄, C₂⁴, C₂²wrC₂, C₂²xD₄, D₄.₈D₄, C₂xC₂²wrC₂, C₂xD₄.₈D₄

Smallest permutation representation of C₂xD₄.₈D₄
►On 32 points

Generators in S₃₂

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

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

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

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

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

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

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

32 conjugacy classes

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

32 irreducible representations

dim

type

image

C₁

C₂

D₄

D₄.₈D₄

kernel

C₂xD₄.₈D₄

C₂xC₄.₁₀D₄

C₂xC₄wrC₂

D₄.₈D₄

C₂xC₄.₄D₄

C₂xC₈:C₂²

C₂x2_-¹⁺⁴

C₂²xC₄

C₂xD₄

C₂xQ₈

C₄oD₄

C₂

# reps

Matrix representation of C₂xD₄.₈D₄ ►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	8	8	0	1
0	0	8	9	16	0

1	15	0	0	0	0
0	16	0	0	0	0
0	0	2	2	0	13
0	0	0	0	13	0
0	0	0	4	0	0
0	0	5	1	15	15

1	15	0	0	0	0
1	16	0	0	0	0
0	0	0	0	1	0
0	0	9	9	0	16
0	0	0	1	0	0
0	0	14	13	9	8

1	0	0	0	0	0
1	16	0	0	0	0
0	0	1	0	0	0
0	0	0	16	0	0
0	0	8	8	0	1
0	0	9	8	1	0

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],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,16,8,8,0,0,1,0,8,9,0,0,0,0,0,16,0,0,0,0,1,0],[1,0,0,0,0,0,15,16,0,0,0,0,0,0,2,0,0,5,0,0,2,0,4,1,0,0,0,13,0,15,0,0,13,0,0,15],[1,1,0,0,0,0,15,16,0,0,0,0,0,0,0,9,0,14,0,0,0,9,1,13,0,0,1,0,0,9,0,0,0,16,0,8],[1,1,0,0,0,0,0,16,0,0,0,0,0,0,1,0,8,9,0,0,0,16,8,8,0,0,0,0,0,1,0,0,0,0,1,0] >;

C₂xD₄.₈D₄ in GAP, Magma, Sage, TeX

C_2\times D_4._8D_4

% in TeX

G:=Group("C2xD4.8D4");

// GroupNames label

G:=SmallGroup(128,1748);

// by ID

G=gap.SmallGroup(128,1748);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,352,2804,1411,718,172,2028]);

// 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=d*b*d^-1=e*b*e=b^-1,d*c*d^-1=b*c,e*c*e=b^-1*c,e*d*e=b^2*d^3>;

// generators/relations

G = C2xD4.8D4 order 128 = 27

Direct product of C2 and D4.8D4

G = C₂xD₄.₈D₄ order 128 = 2⁷

Direct product of C₂ and D₄.₈D₄