C2^4.78D4 - GroupNames

G = C₂⁴.₇₈D₄ order 128 = 2⁷

33^rd non-split extension by C₂⁴ of D₄ acting via D₄/C₂=C₂²

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

Aliases: C₂⁴.₇₈D₄, C₂⁵.₁₀C₂², C₂⁴.₁₇₃C₂³, C₂⁴⋊₁(C₂×C₄), C₂²⋊C₄⋊₃₀D₄, (C₂²×D₄)⋊₇C₄, C₂⁴⋊₃C₄⋊₃C₂, C₂².₄₉(C₄×D₄), C₂²⋊₁(C₂³⋊C₄), C₂³.₅₆₈(C₂×D₄), C₂³.₉D₄⋊₈C₂, C₂².₂₉C₂²≀C₂, C₂³.₁₂₀(C₄○D₄), C₂².₄₇(C₄⋊D₄), C₂³.₁₉₂(C₂²×C₄), (C₂²×D₄).₂₆C₂², C₂³.₁₂₄(C₂²⋊C₄), C₂.₃₇(C₂³.₂₃D₄), C₂².₂₇(C₂².D₄), (C₂×C₂²⋊C₄)⋊₄C₄, (C₂×C₂³⋊C₄)⋊₅C₂, (C₂²×C₄)⋊₁(C₂×C₄), C₂.₂₇(C₂×C₂³⋊C₄), (C₂×C₂²≀C₂).₂C₂, (C₂²×C₂²⋊C₄)⋊₂C₂, (C₂×C₂²⋊C₄).₁₂C₂², C₂².₂₇₃(C₂×C₂²⋊C₄), SmallGroup(128,630)

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

Derived series

C₁ — C₂³ — C₂⁴.₇₈D₄

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂⁴ — C₂⁵ — C₂²×C₂²⋊C₄ — C₂⁴.₇₈D₄

Lower central

C₁ — C₂ — C₂³ — C₂⁴.₇₈D₄

Upper central

C₁ — C₂² — C₂⁵ — C₂⁴.₇₈D₄

Jennings

C₁ — C₂ — C₂⁴ — C₂⁴.₇₈D₄

Generators and relations for C₂⁴.₇₈D₄
G = < a,b,c,d,e,f | a²=b²=c²=d²=e⁴=1, f²=d, faf^-1=ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, bf=fb, ece^-1=fcf^-1=cd=dc, de=ed, df=fd, fef^-1=cde^-1 >

Subgroups: 788 in 299 conjugacy classes, 60 normal (20 characteristic)
C₁, C₂, C₂, C₂, C₄, C₂², C₂², C₂², C₂×C₄, D₄, C₂³, C₂³, C₂³, C₂²⋊C₄, C₂²⋊C₄, C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂⁴, C₂⁴, C₂⁴, C₂³⋊C₄, C₂×C₂²⋊C₄, C₂×C₂²⋊C₄, C₂×C₂²⋊C₄, C₂²≀C₂, C₂³×C₄, C₂²×D₄, C₂²×D₄, C₂⁵, C₂³.₉D₄, C₂⁴⋊₃C₄, C₂×C₂³⋊C₄, C₂²×C₂²⋊C₄, C₂×C₂²≀C₂, C₂⁴.₇₈D₄
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, C₂³, C₂²⋊C₄, C₂²×C₄, C₂×D₄, C₄○D₄, C₂³⋊C₄, C₂×C₂²⋊C₄, C₄×D₄, C₂²≀C₂, C₄⋊D₄, C₂².D₄, C₂³.₂₃D₄, C₂×C₂³⋊C₄, C₂⁴.₇₈D₄

Permutation representations of C₂⁴.₇₈D₄
►On 16 points - transitive group 16T230

Generators in S₁₆

(9 14)(10 15)(11 16)(12 13)

(1 3)(2 4)(5 8)(6 7)(9 14)(10 15)(11 16)(12 13)

(1 5)(2 7)(3 8)(4 6)(9 12)(10 11)(13 14)(15 16)

(1 4)(2 3)(5 6)(7 8)(9 11)(10 12)(13 15)(14 16)

(5 6)(7 8)(9 10 11 12)(13 14 15 16)

(1 14 4 16)(2 11 3 9)(5 15 6 13)(7 12 8 10)

G:=sub<Sym(16)| (9,14)(10,15)(11,16)(12,13), (1,3)(2,4)(5,8)(6,7)(9,14)(10,15)(11,16)(12,13), (1,5)(2,7)(3,8)(4,6)(9,12)(10,11)(13,14)(15,16), (1,4)(2,3)(5,6)(7,8)(9,11)(10,12)(13,15)(14,16), (5,6)(7,8)(9,10,11,12)(13,14,15,16), (1,14,4,16)(2,11,3,9)(5,15,6,13)(7,12,8,10)>;

G:=Group( (9,14)(10,15)(11,16)(12,13), (1,3)(2,4)(5,8)(6,7)(9,14)(10,15)(11,16)(12,13), (1,5)(2,7)(3,8)(4,6)(9,12)(10,11)(13,14)(15,16), (1,4)(2,3)(5,6)(7,8)(9,11)(10,12)(13,15)(14,16), (5,6)(7,8)(9,10,11,12)(13,14,15,16), (1,14,4,16)(2,11,3,9)(5,15,6,13)(7,12,8,10) );

G=PermutationGroup([[(9,14),(10,15),(11,16),(12,13)], [(1,3),(2,4),(5,8),(6,7),(9,14),(10,15),(11,16),(12,13)], [(1,5),(2,7),(3,8),(4,6),(9,12),(10,11),(13,14),(15,16)], [(1,4),(2,3),(5,6),(7,8),(9,11),(10,12),(13,15),(14,16)], [(5,6),(7,8),(9,10,11,12),(13,14,15,16)], [(1,14,4,16),(2,11,3,9),(5,15,6,13),(7,12,8,10)]])

G:=TransitiveGroup(16,230);

►On 16 points - transitive group 16T237

Generators in S₁₆

(5 15)(6 16)(7 13)(8 14)

(1 3)(2 4)(5 15)(6 16)(7 13)(8 14)(9 11)(10 12)

(1 3)(2 10)(4 12)(5 7)(6 16)(8 14)(9 11)(13 15)

(1 11)(2 12)(3 9)(4 10)(5 13)(6 14)(7 15)(8 16)

(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)

(1 13 11 5)(2 6 12 14)(3 7 9 15)(4 16 10 8)

G:=sub<Sym(16)| (5,15)(6,16)(7,13)(8,14), (1,3)(2,4)(5,15)(6,16)(7,13)(8,14)(9,11)(10,12), (1,3)(2,10)(4,12)(5,7)(6,16)(8,14)(9,11)(13,15), (1,11)(2,12)(3,9)(4,10)(5,13)(6,14)(7,15)(8,16), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,13,11,5)(2,6,12,14)(3,7,9,15)(4,16,10,8)>;

G:=Group( (5,15)(6,16)(7,13)(8,14), (1,3)(2,4)(5,15)(6,16)(7,13)(8,14)(9,11)(10,12), (1,3)(2,10)(4,12)(5,7)(6,16)(8,14)(9,11)(13,15), (1,11)(2,12)(3,9)(4,10)(5,13)(6,14)(7,15)(8,16), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,13,11,5)(2,6,12,14)(3,7,9,15)(4,16,10,8) );

G=PermutationGroup([[(5,15),(6,16),(7,13),(8,14)], [(1,3),(2,4),(5,15),(6,16),(7,13),(8,14),(9,11),(10,12)], [(1,3),(2,10),(4,12),(5,7),(6,16),(8,14),(9,11),(13,15)], [(1,11),(2,12),(3,9),(4,10),(5,13),(6,14),(7,15),(8,16)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(1,13,11,5),(2,6,12,14),(3,7,9,15),(4,16,10,8)]])

G:=TransitiveGroup(16,237);

►On 16 points - transitive group 16T249

Generators in S₁₆

(1 4)(2 3)(5 7)(6 8)

(1 4)(2 3)(5 7)(6 8)(9 15)(10 16)(11 13)(12 14)

(1 5)(4 7)(9 11)(13 15)

(1 5)(2 6)(3 8)(4 7)(9 11)(10 12)(13 15)(14 16)

(1 2)(3 4)(5 6)(7 8)(9 10 11 12)(13 14 15 16)

(1 12 5 10)(2 9 6 11)(3 15 8 13)(4 14 7 16)

G:=sub<Sym(16)| (1,4)(2,3)(5,7)(6,8), (1,4)(2,3)(5,7)(6,8)(9,15)(10,16)(11,13)(12,14), (1,5)(4,7)(9,11)(13,15), (1,5)(2,6)(3,8)(4,7)(9,11)(10,12)(13,15)(14,16), (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16), (1,12,5,10)(2,9,6,11)(3,15,8,13)(4,14,7,16)>;

G:=Group( (1,4)(2,3)(5,7)(6,8), (1,4)(2,3)(5,7)(6,8)(9,15)(10,16)(11,13)(12,14), (1,5)(4,7)(9,11)(13,15), (1,5)(2,6)(3,8)(4,7)(9,11)(10,12)(13,15)(14,16), (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16), (1,12,5,10)(2,9,6,11)(3,15,8,13)(4,14,7,16) );

G=PermutationGroup([[(1,4),(2,3),(5,7),(6,8)], [(1,4),(2,3),(5,7),(6,8),(9,15),(10,16),(11,13),(12,14)], [(1,5),(4,7),(9,11),(13,15)], [(1,5),(2,6),(3,8),(4,7),(9,11),(10,12),(13,15),(14,16)], [(1,2),(3,4),(5,6),(7,8),(9,10,11,12),(13,14,15,16)], [(1,12,5,10),(2,9,6,11),(3,15,8,13),(4,14,7,16)]])

G:=TransitiveGroup(16,249);

32 conjugacy classes

class	1	2A	2B	2C	2D	···	2M	2N	2O	2P	4A	···	4H	4I	···	4O
order	1	2	2	2	2	···	2	2	2	2	4	···	4	4	···	4
size	1	1	1	1	2	···	2	4	4	8	4	···	4	8	···	8

32 irreducible representations

dim

type

image

C₁

C₂

C₄

D₄

C₄○D₄

C₂³⋊C₄

kernel

C₂⁴.₇₈D₄

C₂³.₉D₄

C₂⁴⋊₃C₄

C₂×C₂³⋊C₄

C₂²×C₂²⋊C₄

C₂×C₂²≀C₂

C₂×C₂²⋊C₄

C₂²×D₄

C₂²⋊C₄

C₂⁴

C₂³

C₂²

# reps

Matrix representation of C₂⁴.₇₈D₄ ►in GL₆(𝔽₅)

4	0	0	0	0	0
0	1	0	0	0	0
0	0	4	0	0	0
0	0	0	4	0	0
0	0	0	0	4	0
0	0	0	0	0	4

4	0	0	0	0	0
0	4	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

4	0	0	0	0	0
0	4	0	0	0	0
0	0	0	1	0	0
0	0	1	0	0	0
0	0	0	0	0	1
0	0	0	0	1	0

1	0	0	0	0	0
0	1	0	0	0	0
0	0	4	0	0	0
0	0	0	4	0	0
0	0	0	0	4	0
0	0	0	0	0	4

2	0	0	0	0	0
0	2	0	0	0	0
0	0	0	4	0	0
0	0	1	0	0	0
0	0	0	0	4	0
0	0	0	0	0	1

0	1	0	0	0	0
1	0	0	0	0	0
0	0	0	0	4	0
0	0	0	0	0	1
0	0	1	0	0	0
0	0	0	4	0	0

G:=sub<GL(6,GF(5))| [4,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[2,0,0,0,0,0,0,2,0,0,0,0,0,0,0,1,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,4,0,0,0,0,0,0,1,0,0] >;

C₂⁴.₇₈D₄ in GAP, Magma, Sage, TeX

C_2^4._{78}D_4

% in TeX

G:=Group("C2^4.78D4");

// GroupNames label

G:=SmallGroup(128,630);

// by ID

G=gap.SmallGroup(128,630);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,2019,1018,2028]);

// Polycyclic

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

// generators/relations

G = C24.78D4 order 128 = 27

33rd non-split extension by C24 of D4 acting via D4/C2=C22

G = C₂⁴.₇₈D₄ order 128 = 2⁷

33^rd non-split extension by C₂⁴ of D₄ acting via D₄/C₂=C₂²