C2^3.9D4 - GroupNames

G = C₂³.₉D₄ order 64 = 2⁶

2^nd non-split extension by C₂³ of D₄ acting via D₄/C₂=C₂²

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

Aliases: C₂³.₉D₄, C₂³.₂Q₈, C₂².₁C₄², C₂⁴.₁C₂², C₂²⋊C₄⋊₃C₄, (C₂²×C₄)⋊₁C₄, C₂³.₅(C₂×C₄), C₂.₃(C₂³⋊C₄), C₂².₄(C₄⋊C₄), C₂².₇(C₂²⋊C₄), C₂.₇(C₂.C₄²), (C₂×C₂²⋊C₄).₂C₂, SmallGroup(64,23)

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

Derived series

C₁ — C₂² — C₂³.₉D₄

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂⁴ — C₂×C₂²⋊C₄ — C₂³.₉D₄

Lower central

C₁ — C₂ — C₂² — C₂³.₉D₄

Upper central

C₁ — C₂² — C₂⁴ — C₂³.₉D₄

Jennings

C₁ — C₂ — C₂⁴ — C₂³.₉D₄

Generators and relations for C₂³.₉D₄
G = < a,b,c,d,e | a²=b²=c²=d⁴=1, e²=abc, ab=ba, eae^-1=ac=ca, ad=da, dbd^-1=ebe^-1=bc=cb, cd=dc, ce=ec, ede^-1=acd^-1 >

Subgroups: 149 in 71 conjugacy classes, 29 normal (11 characteristic)
C₁, C₂, C₂, C₂, C₄, C₂², C₂², C₂², C₂×C₄, C₂³, C₂³, C₂³, C₂²⋊C₄, C₂²⋊C₄, C₂²×C₄, C₂²×C₄, C₂⁴, C₂×C₂²⋊C₄, C₂×C₂²⋊C₄, C₂³.₉D₄
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, Q₈, C₄², C₂²⋊C₄, C₄⋊C₄, C₂.C₄², C₂³⋊C₄, C₂³.₉D₄

Character table of C₂³.₉D₄

class	1	2A	2B	2C	2D	2E	2F	2G	2H	2I	4A	4B	4C	4D	4E	4F	4G	4H	4I	4J	4K	4L
size	1	1	1	1	2	2	2	2	2	2	4	4	4	4	4	4	4	4	4	4	4	4
ρ₁	1	1	1	1	1	1	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	-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	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	-1	1	1	1	-1	1	linear of order 2
ρ₅	1	-1	1	-1	1	-1	1	1	-1	-1	-i	i	i	-i	i	i	-i	1	1	-1	-i	-1	linear of order 4
ρ₆	1	-1	1	-1	1	-1	1	1	-1	-1	i	-i	-i	i	i	i	-i	-1	-1	1	-i	1	linear of order 4
ρ₇	1	1	1	1	-1	-1	-1	1	1	-1	1	1	-1	-1	i	-i	-i	i	-i	-i	i	i	linear of order 4
ρ₈	1	-1	1	-1	1	-1	1	1	-1	-1	i	-i	-i	i	-i	-i	i	1	1	-1	i	-1	linear of order 4
ρ₉	1	-1	1	-1	1	-1	1	1	-1	-1	-i	i	i	-i	-i	-i	i	-1	-1	1	i	1	linear of order 4
ρ₁₀	1	1	1	1	-1	-1	-1	1	1	-1	1	1	-1	-1	-i	i	i	-i	i	i	-i	-i	linear of order 4
ρ₁₁	1	-1	1	-1	-1	1	-1	1	-1	1	i	-i	i	-i	-1	1	-1	-i	i	-i	1	i	linear of order 4
ρ₁₂	1	-1	1	-1	-1	1	-1	1	-1	1	-i	i	-i	i	-1	1	-1	i	-i	i	1	-i	linear of order 4
ρ₁₃	1	1	1	1	-1	-1	-1	1	1	-1	-1	-1	1	1	i	-i	-i	-i	i	i	i	-i	linear of order 4
ρ₁₄	1	-1	1	-1	-1	1	-1	1	-1	1	-i	i	-i	i	1	-1	1	-i	i	-i	-1	i	linear of order 4
ρ₁₅	1	-1	1	-1	-1	1	-1	1	-1	1	i	-i	i	-i	1	-1	1	i	-i	i	-1	-i	linear of order 4
ρ₁₆	1	1	1	1	-1	-1	-1	1	1	-1	-1	-1	1	1	-i	i	i	i	-i	-i	-i	i	linear of order 4
ρ₁₇	2	2	2	2	-2	2	2	-2	-2	-2	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₈	2	-2	2	-2	2	2	-2	-2	2	-2	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₉	2	2	2	2	2	-2	-2	-2	-2	2	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₂₀	2	-2	2	-2	-2	-2	2	-2	2	2	0	0	0	0	0	0	0	0	0	0	0	0	symplectic lifted from Q₈, Schur index 2
ρ₂₁	4	4	-4	-4	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from C₂³⋊C₄
ρ₂₂	4	-4	-4	4	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from C₂³⋊C₄

Permutation representations of C₂³.₉D₄
►On 16 points - transitive group 16T77

Generators in S₁₆

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

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

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

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

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

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

G:=TransitiveGroup(16,77);

►On 16 points - transitive group 16T88

Generators in S₁₆

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

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

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

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

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

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

G:=TransitiveGroup(16,88);

►On 16 points - transitive group 16T91

Generators in S₁₆

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

(2 10)(4 12)(5 13)(7 15)

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

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

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

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

G:=TransitiveGroup(16,91);

►On 16 points - transitive group 16T96

Generators in S₁₆

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

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

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

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

(1 5 15 12)(3 7 13 10)(6 9)(8 11)

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

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

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

G:=TransitiveGroup(16,96);

C₂³.₉D₄ is a maximal subgroup of
C₂⁴.₁₆₂C₂³ C₄×C₂³⋊C₄ C₂⁴.₁₆₇C₂³ C₂⁴.₁₆₉C₂³ C₂⁴.C₂³ C₂⁴.₁₇₄C₂³ C₂⁴.₁₇₆C₂³ C₂⁴⋊D₄ C₂⁴⋊₂Q₈ C₂⁴.₁₈₀C₂³ C₂⁴⋊Q₈ C₂⁴.₁₈₂C₂³ C₂⁴.₄C₂³ (C₂²×F₅)⋊C₄ C₂²⋊F₅⋊C₄
C₂⁴.D_2p: C₂⁴.D₄ C₂³.₄D₈ C₂³.Q₁₆ C₂⁴.₄D₄ C₂⁴.₆₈D₄ C₂⁴.₂₂D₄ C₂⁵.C₂² C₂⁴.₂₆D₄ ...
C₂³.₉D₄ is a maximal quotient of
C₂³.₁₉C₄² C₂³.₂₁C₄² C₂⁴.₂Q₈ C₂⁴.₃Q₈ C₂³.C₄² C₂³.₈C₄² C₂³.₂C₄² C₂³.₃C₄² (C₂×Q₈).Q₈ (C₂²×C₈)⋊C₄ (C₂²×F₅)⋊C₄ C₂²⋊F₅⋊C₄
C₂⁴.D_2p: C₂⁴.₄₆D₄ C₂³.₈D₈ C₂³.₃₀D₈ C₂⁴.₄₈D₄ C₂⁴.₄Q₈ C₂⁴.₅D₄ C₂⁴.₆D₄ C₂⁴.₁₂D₆ ...

Matrix representation of C₂³.₉D₄ ►in GL₆(𝔽₅)

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

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	4	0
0	0	4	1	0	4

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

4	4	0	0	0	0
2	1	0	0	0	0
0	0	0	0	1	0
0	0	4	1	1	3
0	0	4	0	0	0
0	0	4	1	0	4

3	3	0	0	0	0
0	2	0	0	0	0
0	0	0	0	4	0
0	0	4	1	1	3
0	0	0	4	0	0
0	0	4	0	1	4

G:=sub<GL(6,GF(5))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,4,1,0,0,0,4,0,4,0,0,0,0,0,4,0,0,0,0,0,2,1],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,4,0,0,0,1,0,1,0,0,0,0,4,0,0,0,0,0,0,4],[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],[4,2,0,0,0,0,4,1,0,0,0,0,0,0,0,4,4,4,0,0,0,1,0,1,0,0,1,1,0,0,0,0,0,3,0,4],[3,0,0,0,0,0,3,2,0,0,0,0,0,0,0,4,0,4,0,0,0,1,4,0,0,0,4,1,0,1,0,0,0,3,0,4] >;

C₂³.₉D₄ in GAP, Magma, Sage, TeX

C_2^3._9D_4

% in TeX

G:=Group("C2^3.9D4");

// GroupNames label

G:=SmallGroup(64,23);

// by ID

G=gap.SmallGroup(64,23);

# by ID

G:=PCGroup([6,-2,2,-2,2,2,-2,48,73,103,650,489]);

// Polycyclic

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

// generators/relations

Export

Character table of C₂³.₉D₄ in TeX

G = C23.9D4 order 64 = 26

2nd non-split extension by C23 of D4 acting via D4/C2=C22

G = C₂³.₉D₄ order 64 = 2⁶

2^nd non-split extension by C₂³ of D₄ acting via D₄/C₂=C₂²