C2^4:3C4 - GroupNames

G = C₂⁴⋊₃C₄ order 64 = 2⁶

1^st semidirect product of C₂⁴ and C₄ acting via C₄/C₂=C₂

p-group, metabelian, nilpotent (class 2), monomial

Aliases: C₂⁴⋊₃C₄, C₂⁵.₁C₂, C₂³.₃₃D₄, C₂³.₅₅C₂³, C₂⁴.₂₃C₂², C₂.₁C₂²≀C₂, C₂³.₂₄(C₂×C₄), (C₂²×C₄)⋊₁C₂², C₂².₂₉(C₂×D₄), C₂²⋊₂(C₂²⋊C₄), C₂².₂₈(C₂²×C₄), (C₂×C₂²⋊C₄)⋊₁C₂, C₂.₄(C₂×C₂²⋊C₄), SmallGroup(64,60)

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

Derived series

C₁ — C₂² — C₂⁴⋊₃C₄

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂⁴ — C₂⁵ — C₂⁴⋊₃C₄

Lower central

C₁ — C₂² — C₂⁴⋊₃C₄

Upper central

C₁ — C₂³ — C₂⁴⋊₃C₄

Jennings

C₁ — C₂³ — C₂⁴⋊₃C₄

Generators and relations for C₂⁴⋊₃C₄
G = < a,b,c,d,e | a²=b²=c²=d²=e⁴=1, ab=ba, eae^-1=ac=ca, ad=da, bc=cb, ebe^-1=bd=db, cd=dc, ce=ec, de=ed >

Subgroups: 449 in 253 conjugacy classes, 65 normal (5 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₂², C₂×C₄, C₂³, C₂³, C₂³, C₂²⋊C₄, C₂²×C₄, C₂⁴, C₂⁴, C₂×C₂²⋊C₄, C₂⁵, C₂⁴⋊₃C₄
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, C₂³, C₂²⋊C₄, C₂²×C₄, C₂×D₄, C₂×C₂²⋊C₄, C₂²≀C₂, C₂⁴⋊₃C₄

Character table of C₂⁴⋊₃C₄

class	1	2A	2B	2C	2D	2E	2F	2G	2H	2I	2J	2K	2L	2M	2N	2O	2P	2Q	2R	2S	4A	4B	4C	4D	4E	4F	4G	4H
size	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	2	2	2	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	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	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	-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	-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	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	-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	-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	1	1	1	1	-1	-1	-1	-1	-1	-1	-1	-1	linear of order 2
ρ₉	1	-1	1	-1	1	-1	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
ρ₁₀	1	-1	1	-1	1	-1	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	-1	-1	1	-1	1	-1	1	1	-1	1	i	-i	-i	i	-i	-i	i	i	linear of order 4
ρ₁₂	1	-1	1	-1	1	-1	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
ρ₁₃	1	-1	1	-1	1	-1	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	-1	1	-1	1	-1	1	-1	1	-1	1	i	-i	i	-i	i	-i	i	-i	linear of order 4
ρ₁₅	1	-1	1	-1	1	-1	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
ρ₁₆	1	-1	1	-1	1	-1	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	0	0	-2	-2	2	0	0	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	0	2	0	0	0	0	-2	-2	2	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	0	0	2	-2	-2	0	0	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	0	0	-2	2	2	0	0	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₂₁	2	2	-2	-2	-2	-2	2	2	0	0	2	0	0	0	0	0	0	2	-2	-2	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₂₂	2	-2	-2	2	-2	2	2	-2	0	0	-2	0	0	0	0	0	0	2	2	-2	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₂₃	2	2	-2	2	2	-2	-2	-2	0	-2	0	0	0	0	2	2	-2	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₂₄	2	2	2	-2	-2	2	-2	-2	-2	0	0	2	2	-2	0	0	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₂₅	2	-2	-2	-2	2	2	-2	2	0	2	0	0	0	0	2	-2	-2	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₂₆	2	-2	-2	2	-2	2	2	-2	0	0	2	0	0	0	0	0	0	-2	-2	2	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₂₇	2	-2	-2	-2	2	2	-2	2	0	-2	0	0	0	0	-2	2	2	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₂₈	2	2	-2	-2	-2	-2	2	2	0	0	-2	0	0	0	0	0	0	-2	2	2	0	0	0	0	0	0	0	0	orthogonal lifted from D₄

Permutation representations of C₂⁴⋊₃C₄
►On 16 points - transitive group 16T79

Generators in S₁₆

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

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

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

(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)

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

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

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

G:=TransitiveGroup(16,79);

C₂⁴⋊₃C₄ is a maximal subgroup of
C₂⁴⋊C₈ C₂⁴⋊D₄ C₂⁴⋊₂Q₈ C₂⁵.₈₅C₂² C₄×C₂²≀C₂ C₂³.₁₉₁C₂⁴ C₂³.₁₉₄C₂⁴ C₂³.₂₀₃C₂⁴ D₄×C₂²⋊C₄ C₂³.₂₂₄C₂⁴ C₂³.₂₅₇C₂⁴ C₂⁴⋊₇D₄ C₂³.₃₀₈C₂⁴ C₂⁴⋊₈D₄ C₂³.₃₁₁C₂⁴ C₂³.₃₁₈C₂⁴ C₂³.₃₃₅C₂⁴ C₂⁴⋊₄Q₈ C₂³.₃₇₂C₂⁴ C₂³.₃₈₀C₂⁴ C₂³.₃₈₂C₂⁴ C₂³.₄₃₄C₂⁴ C₂³.₄₆₁C₂⁴ C₂⁴⋊₉D₄ C₂⁴⋊₅Q₈ C₂³.₅₆₈C₂⁴ C₂³.₅₇₀C₂⁴ C₂³.₅₇₈C₂⁴ C₂³.₅₈₄C₂⁴ C₂³.₅₈₅C₂⁴ C₂³.₅₉₇C₂⁴ C₂³.₆₃₅C₂⁴ C₂³.₆₃₆C₂⁴ C₂⁴⋊₁₁D₄ C₂⁴⋊₆Q₈ C₂⁴.₁₅Q₈ C₂³≀C₂ C₂⁴⋊₈Q₈ C₂⁴⋊C₁₂ C₂⁴⋊₄Dic₃ C₂⁴⋊₄F₅ C₂⁵.D₅
C₂⁴.D_2p: C₂⁴.₄Q₈ C₂⁵⋊C₄ C₂⁴.₆₈D₄ C₂⁵.C₂² C₂⁴.₇₈D₄ C₂⁴.₉₀D₄ C₂⁴.₉₄D₄ C₂⁴.₉₆D₄ ...
C₂⁴⋊₃C₄ is a maximal quotient of
C₂⁴.₆₃₆C₂³ C₂⁴⋊₃C₈ C₂⁴.₅₁(C₂×C₄) C₂⁴.₁₆₅C₂³ C₂⁵.C₄ C₄.C₂²≀C₂ (C₂³×C₄).C₄ 2₊¹⁺⁴.₂C₄ 2_-¹⁺⁴⋊₂C₄ 2₊¹⁺⁴⋊₄C₄ C₄○D₄.D₄ (C₂²×Q₈)⋊C₄ C₂⁴⋊₄F₅
C₂⁴.D_2p: C₂⁴.₁₇Q₈ C₂⁴.₅₀D₄ C₂⁵⋊C₄ C₂³.₃₅D₈ C₂⁴.₁₅₅D₄ C₂⁴.₆₅D₄ C₂⁴.₆₆D₄ 2₊¹⁺⁴⋊₂C₄ ...

Matrix representation of C₂⁴⋊₃C₄ ►in GL₅(𝔽₅)

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

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

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

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

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

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

C₂⁴⋊₃C₄ in GAP, Magma, Sage, TeX

C_2^4\rtimes_3C_4

% in TeX

G:=Group("C2^4:3C4");

// GroupNames label

G:=SmallGroup(64,60);

// by ID

G=gap.SmallGroup(64,60);

# by ID

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

// Polycyclic

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

// generators/relations

Export

Character table of C₂⁴⋊₃C₄ in TeX

G = C24⋊3C4 order 64 = 26

1st semidirect product of C24 and C4 acting via C4/C2=C2

G = C₂⁴⋊₃C₄ order 64 = 2⁶

1^st semidirect product of C₂⁴ and C₄ acting via C₄/C₂=C₂