C2^4.4C4 - GroupNames

G = C₂⁴.₄C₄ order 64 = 2⁶

2^nd non-split extension by C₂⁴ of C₄ acting via C₄/C₂=C₂

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

Aliases: C₂⁴.₄C₄, C₂²⋊₂M₄(2), (C₂×C₈)⋊₇C₂², C₄.₆₇(C₂×D₄), C₂²⋊C₈⋊₁₁C₂, (C₂×C₄).₁₁₇D₄, (C₂³×C₄).₇C₂, (C₂×M₄(2))⋊₆C₂, (C₂²×C₄).₁₃C₄, C₂³.₂₉(C₂×C₄), C₂.₆(C₂×M₄(2)), C₄.₂₃(C₂²⋊C₄), (C₂×C₄).₁₄₅C₂³, (C₂²×C₄).₉₂C₂², C₂².₄₁(C₂²×C₄), C₂².₁₆(C₂²⋊C₄), (C₂×C₄).₇₁(C₂×C₄), C₂.₁₀(C₂×C₂²⋊C₄), SmallGroup(64,88)

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

Derived series

C₁ — C₂² — C₂⁴.₄C₄

Chief series

C₁ — C₂ — C₄ — C₂×C₄ — C₂²×C₄ — C₂³×C₄ — C₂⁴.₄C₄

Lower central

C₁ — C₂² — C₂⁴.₄C₄

Upper central

C₁ — C₂×C₄ — C₂⁴.₄C₄

Jennings

C₁ — C₂ — C₂ — C₂×C₄ — C₂⁴.₄C₄

Generators and relations for C₂⁴.₄C₄
G = < a,b,c,d,e | a²=b²=c²=d²=1, e⁴=d, ab=ba, eae^-1=ac=ca, ad=da, bc=cb, ebe^-1=bd=db, cd=dc, ce=ec, de=ed >

Subgroups: 145 in 95 conjugacy classes, 45 normal (11 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₂², C₂², C₂², C₈, C₂×C₄, C₂×C₄, C₂×C₄, C₂³, C₂³, C₂³, C₂×C₈, M₄(2), C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂⁴, C₂²⋊C₈, C₂×M₄(2), C₂³×C₄, C₂⁴.₄C₄
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, C₂³, C₂²⋊C₄, M₄(2), C₂²×C₄, C₂×D₄, C₂×C₂²⋊C₄, C₂×M₄(2), C₂⁴.₄C₄

Character table of C₂⁴.₄C₄

class	1	2A	2B	2C	2D	2E	2F	2G	2H	2I	4A	4B	4C	4D	4E	4F	4G	4H	4I	4J	8A	8B	8C	8D	8E	8F	8G	8H
size	1	1	1	1	2	2	2	2	2	2	1	1	1	1	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	0	0	0	0	-2	-2	2	2	-2	0	0	-2	2	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₈	2	-2	-2	2	2	0	0	0	0	-2	2	-2	-2	2	0	0	2	-2	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₉	2	-2	-2	2	-2	0	0	0	0	2	-2	2	2	-2	0	0	2	-2	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₂₀	2	-2	-2	2	-2	0	0	0	0	2	2	-2	-2	2	0	0	-2	2	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₂₁	2	-2	2	-2	0	0	-2	0	2	0	2i	2i	-2i	-2i	-2i	2i	0	0	0	0	0	0	0	0	0	0	0	0	complex lifted from M₄(2)
ρ₂₂	2	-2	2	-2	0	0	2	0	-2	0	2i	2i	-2i	-2i	2i	-2i	0	0	0	0	0	0	0	0	0	0	0	0	complex lifted from M₄(2)
ρ₂₃	2	2	-2	-2	0	-2	0	2	0	0	-2i	2i	-2i	2i	0	0	0	0	-2i	2i	0	0	0	0	0	0	0	0	complex lifted from M₄(2)
ρ₂₄	2	-2	2	-2	0	0	-2	0	2	0	-2i	-2i	2i	2i	2i	-2i	0	0	0	0	0	0	0	0	0	0	0	0	complex lifted from M₄(2)
ρ₂₅	2	-2	2	-2	0	0	2	0	-2	0	-2i	-2i	2i	2i	-2i	2i	0	0	0	0	0	0	0	0	0	0	0	0	complex lifted from M₄(2)
ρ₂₆	2	2	-2	-2	0	-2	0	2	0	0	2i	-2i	2i	-2i	0	0	0	0	2i	-2i	0	0	0	0	0	0	0	0	complex lifted from M₄(2)
ρ₂₇	2	2	-2	-2	0	2	0	-2	0	0	2i	-2i	2i	-2i	0	0	0	0	-2i	2i	0	0	0	0	0	0	0	0	complex lifted from M₄(2)
ρ₂₈	2	2	-2	-2	0	2	0	-2	0	0	-2i	2i	-2i	2i	0	0	0	0	2i	-2i	0	0	0	0	0	0	0	0	complex lifted from M₄(2)

Permutation representations of C₂⁴.₄C₄
►On 16 points - transitive group 16T95

Generators in S₁₆

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

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

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

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

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

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

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

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

G:=TransitiveGroup(16,95);

C₂⁴.₄C₄ is a maximal subgroup of
C₂⁴.₄₆D₄ C₂⁴.₂Q₈ C₂⁴.₄₈D₄ C₂⁴.₃Q₈ C₂³.C₄² C₂³.₈C₄² C₂⁴.C₈ C₂⁵.₃C₄ C₂³⋊C₈⋊C₂ C₂⁴.(C₂×C₄) C₂⁴.₄₅(C₂×C₄) C₂⁴.₁₅₀D₄ C₂⁴.₅₄D₄ C₂⁴.₅₅D₄ C₂⁴.₅₆D₄ C₂⁴.₅₇D₄ C₂⁴.₅₈D₄ C₂⁵.C₄ C₄.C₂²≀C₂ (C₂³×C₄).C₄ C₂⁴.₆₆D₄ C₂⁴.₁₉Q₈ C₂⁴.₉Q₈ (C₂²×C₄).₂₇₅D₄ (C₂²×C₄).₂₇₆D₄ C₂⁴.₇₀D₄ C₂⁴.₁₀Q₈ (C₂×C₈)⋊D₄ (C₂×C₄)≀C₂ C₄²⋊₇D₄ M₄(2)⋊₂₀D₄ M₄(2).₄₅D₄ C₄²⋊₂D₄ C₂⁴.₁₁Q₈ C₂⁴.₇₃(C₂×C₄) D₄○(C₂²⋊C₈) C₄².₂₅₉C₂³ C₄².₂₆₂C₂³ C₄².₂₆₄C₂³ C₄².₂₆₅C₂³ D₄×M₄(2) C₂³⋊₃M₄(2) C₄².₂₉₇C₂³ C₄².₂₉₉C₂³ C₂⁴.₁₇₇D₄ C₂⁴.₁₇₈D₄ C₂⁴.₁₀₄D₄ C₂⁴.₁₀₅D₄ C₂⁴.₁₀₆D₄ C₂⁴.₁₈₃D₄ C₂⁴.₁₁₆D₄ C₂⁴.₁₁₇D₄ C₂⁴.₁₁₈D₄ C₂⁴.₁₂₅D₄ C₂⁴.₁₂₆D₄ C₂⁴.₁₂₇D₄ C₂⁴.₁₂₈D₄ C₂⁴.₁₂₉D₄ C₂⁴.₁₃₀D₄ A₄⋊M₄(2)
D_2p⋊M₄(2): D₄⋊₇M₄(2) D₆⋊M₄(2) D₆⋊₆M₄(2) D₁₀⋊₇M₄(2) D₁₀⋊₈M₄(2) D₁₀⋊₉M₄(2) D₁₀⋊₁₀M₄(2) D₁₄⋊M₄(2) ...
(C₂×C_2p)⋊M₄(2): (C₂×C₄)⋊M₄(2) C₂³⋊M₄(2) C₄².₆₇₇C₂³ C₄².₆₉₃C₂³ C₂⁴.₆Dic₃ C₂⁴.₄Dic₅ C₂⁴.₄F₅ C₂⁴.₄Dic₇ ...
C₂⁴.₄C₄ is a maximal quotient of
C₂⁵.₃C₄ C₄².₄₂D₄ C₄².₄₃D₄ C₄².₄₄D₄ Q₈⋊M₄(2) C₄².₃₇₄D₄ Q₈⋊₅M₄(2) C₄².₃₇₈D₄ C₂⁴⋊₃C₈ C₄².₄₂₅D₄ C₂³.₃₂M₄(2) C₄².₁₀₉D₄ C₄².₁₂₀D₄
D_2p⋊M₄(2): D₄⋊M₄(2) D₄⋊₄M₄(2) D₄⋊₅M₄(2) D₆⋊M₄(2) D₆⋊₆M₄(2) D₁₀⋊₇M₄(2) D₁₀⋊₈M₄(2) D₁₀⋊₉M₄(2) ...
(C₂×C_2p)⋊M₄(2): (C₂×C₄)⋊M₄(2) C₂³⋊M₄(2) C₂³.₂₈C₄² C₂³⋊₂M₄(2) C₂⁴.₆Dic₃ C₂⁴.₄Dic₅ C₂⁴.₄F₅ C₂⁴.₄Dic₇ ...

Matrix representation of C₂⁴.₄C₄ ►in GL₄(𝔽₁₇) generated by

1	0	0	0
13	16	0	0
0	0	1	0
0	0	0	1

1	0	0	0
13	16	0	0
0	0	16	0
0	0	0	1

16	0	0	0
0	16	0	0
0	0	1	0
0	0	0	1

16	0	0	0
0	16	0	0
0	0	16	0
0	0	0	16

13	15	0	0
10	4	0	0
0	0	0	1
0	0	4	0

G:=sub<GL(4,GF(17))| [1,13,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[1,13,0,0,0,16,0,0,0,0,16,0,0,0,0,1],[16,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[13,10,0,0,15,4,0,0,0,0,0,4,0,0,1,0] >;

C₂⁴.₄C₄ in GAP, Magma, Sage, TeX

C_2^4._4C_4

% in TeX

G:=Group("C2^4.4C4");

// GroupNames label

G:=SmallGroup(64,88);

// by ID

G=gap.SmallGroup(64,88);

# by ID

G:=PCGroup([6,-2,2,2,-2,2,-2,96,121,650,88]);

// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^2=1,e^4=d,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 C₂⁴.₄C₄ in TeX

G = C24.4C4 order 64 = 26

2nd non-split extension by C24 of C4 acting via C4/C2=C2

G = C₂⁴.₄C₄ order 64 = 2⁶

2^nd non-split extension by C₂⁴ of C₄ acting via C₄/C₂=C₂