C2^3.7D4 - GroupNames

G = C₂³.₇D₄ order 64 = 2⁶

7^th non-split extension by C₂³ of D₄ acting faithfully

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

Aliases: C₂³.₇D₄, C₂³.₄C₂³, 2₊¹⁺⁴.₂C₂, (C₂×C₄).₇D₄, C₂³⋊C₄⋊₄C₂, C₂.₁₉C₂²≀C₂, C₂²⋊C₄⋊₂C₂², (C₂²×C₄)⋊₂C₂², C₂².₁₇(C₂×D₄), (C₂×D₄).₁₀C₂², C₂².D₄⋊₁C₂, SmallGroup(64,139)

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

Derived series

C₁ — C₂³ — C₂³.₇D₄

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂×D₄ — 2₊¹⁺⁴ — C₂³.₇D₄

Lower central

C₁ — C₂ — C₂³ — C₂³.₇D₄

Upper central

C₁ — C₂ — C₂³ — C₂³.₇D₄

Jennings

C₁ — C₂ — C₂³ — C₂³.₇D₄

Generators and relations for C₂³.₇D₄
G = < a,b,c,d,e | a²=b²=c²=d⁴=1, e²=c, ab=ba, ac=ca, dad^-1=eae^-1=abc, dbd^-1=bc=cb, be=eb, cd=dc, ce=ec, ede^-1=cd^-1 >

Subgroups: 161 in 80 conjugacy classes, 27 normal (6 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₂³, C₂³, C₂²⋊C₄, C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₄○D₄, C₂³⋊C₄, C₂².D₄, 2₊¹⁺⁴, C₂³.₇D₄
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₂²≀C₂, C₂³.₇D₄

Character table of C₂³.₇D₄

class	1	2A	2B	2C	2D	2E	2F	2G	4A	4B	4C	4D	4E	4F	4G	4H
size	1	1	2	2	2	4	4	4	4	4	4	4	4	8	8	8
ρ₁	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	trivial
ρ₂	1	1	1	1	1	-1	1	-1	1	1	-1	-1	1	-1	-1	1	linear of order 2
ρ₃	1	1	1	1	1	-1	-1	1	-1	-1	1	-1	-1	1	-1	1	linear of order 2
ρ₄	1	1	1	1	1	1	-1	-1	-1	-1	-1	1	-1	-1	1	1	linear of order 2
ρ₅	1	1	1	1	1	1	1	1	-1	1	1	1	-1	-1	-1	-1	linear of order 2
ρ₆	1	1	1	1	1	-1	1	-1	-1	1	-1	-1	-1	1	1	-1	linear of order 2
ρ₇	1	1	1	1	1	-1	-1	1	1	-1	1	-1	1	-1	1	-1	linear of order 2
ρ₈	1	1	1	1	1	1	-1	-1	1	-1	-1	1	1	1	-1	-1	linear of order 2
ρ₉	2	2	-2	-2	2	0	0	2	0	0	-2	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₀	2	2	2	-2	-2	0	-2	0	0	2	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₁	2	2	-2	2	-2	2	0	0	0	0	0	-2	0	0	0	0	orthogonal lifted from D₄
ρ₁₂	2	2	-2	2	-2	-2	0	0	0	0	0	2	0	0	0	0	orthogonal lifted from D₄
ρ₁₃	2	2	-2	-2	2	0	0	-2	0	0	2	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₄	2	2	2	-2	-2	0	2	0	0	-2	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₅	4	-4	0	0	0	0	0	0	-2i	0	0	0	2i	0	0	0	complex faithful
ρ₁₆	4	-4	0	0	0	0	0	0	2i	0	0	0	-2i	0	0	0	complex faithful

Permutation representations of C₂³.₇D₄
►On 16 points - transitive group 16T146

Generators in S₁₆

(2 6)(3 9)(4 14)(7 13)(8 10)(12 16)

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

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

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

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

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

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

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

G:=TransitiveGroup(16,146);

►On 16 points - transitive group 16T165

Generators in S₁₆

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

(2 10)(4 12)(6 16)(8 14)

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

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

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

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

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

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

G:=TransitiveGroup(16,165);

►On 16 points - transitive group 16T173

Generators in S₁₆

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

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

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

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

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

G:=TransitiveGroup(16,173);

►On 16 points - transitive group 16T177

Generators in S₁₆

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

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

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

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

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

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

G:=TransitiveGroup(16,177);

C₂³.₇D₄ is a maximal subgroup of
C₄².₁₄D₄ C₂³.₇C₂⁴ C₂⁴⋊C₂³ 2₊¹⁺⁴.C₆ C₂³.S₄
C₂²⋊C₄⋊D_2p: C₄²⋊₄D₄ C₄²⋊₆D₄ C₂³.₅D₁₂ C₂²⋊C₄⋊D₆ C₂³.₅D₂₀ C₂²⋊C₄⋊D₁₀ C₂³.₅D₂₈ C₂²⋊C₄⋊D₁₄ ...
(C₂×D₄).D_2p: C₄².₁₃D₄ C₂³.₁₀C₂⁴ 2₊¹⁺⁴.₅S₃ 2₊¹⁺⁴.₂D₅ 2₊¹⁺⁴.₂D₇ ...
C₂³.₇D₄ is a maximal quotient of
C₂⁴.₁₄D₄ (C₂×C₄).SD₁₆ C₂⁴.₁₇D₄ C₄⋊C₄.₂₀D₄ C₂⁴.₂₂D₄ C₂⁴.₁₈₀C₂³ C₂⁴.₁₈₂C₂³
C₂³.D_4p: C₂³.₅D₈ C₂³.₅D₁₂ C₂³.₅D₂₀ C₂³.₅D₂₈ ...
C₂²⋊C₄⋊D_2p: C₂⁴.₃₁D₄ C₂⁴.₃₃D₄ C₂²⋊C₄⋊D₆ C₂²⋊C₄⋊D₁₀ C₂²⋊C₄⋊D₁₄ ...
(C₂×D₄).D_2p: C₄⋊C₄.₁₂D₄ (C₂×C₄).₅D₈ C₂⁴.₁₅D₄ C₂⁴.₁₆D₄ C₄⋊C₄.₁₈D₄ C₄⋊C₄.₁₉D₄ C₂⁴.₁₈D₄ 2₊¹⁺⁴⋊₂C₄ ...

Matrix representation of C₂³.₇D₄ ►in GL₄(𝔽₅) generated by

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

0	0	0	1
0	0	1	0
0	1	0	0
1	0	0	0

4	0	0	0
0	4	0	0
0	0	4	0
0	0	0	4

4	4	4	1
1	1	4	1
4	1	4	4
4	1	1	1

4	4	4	1
4	4	1	4
4	1	4	4
1	4	4	4

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

C₂³.₇D₄ in GAP, Magma, Sage, TeX

C_2^3._7D_4

% in TeX

G:=Group("C2^3.7D4");

// GroupNames label

G:=SmallGroup(64,139);

// by ID

G=gap.SmallGroup(64,139);

# by ID

G:=PCGroup([6,-2,2,2,-2,2,-2,192,121,362,255,730]);

// Polycyclic

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

// generators/relations

Export

Character table of C₂³.₇D₄ in TeX

G = C23.7D4 order 64 = 26

7th non-split extension by C23 of D4 acting faithfully

G = C₂³.₇D₄ order 64 = 2⁶

7^th non-split extension by C₂³ of D₄ acting faithfully