C2^5:C4 - GroupNames

G = C₂⁵⋊C₄ order 128 = 2⁷

1^st semidirect product of C₂⁵ and C₄ acting faithfully

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

Aliases: C₂⁵⋊₁C₄, C₂⁴.₆₄D₄, C₂⁵.₇C₂², C₂⁴.₁₆₄C₂³, (C₂³×C₄)⋊₆C₄, C₂⁴⋊₃C₄⋊₁C₂, (C₂×D₄).₂₅₉D₄, (C₂²×D₄)⋊₁₅C₄, (D₄×C₂³).₂C₂, C₂²⋊₂(C₂³⋊C₄), C₂³⋊₃(C₂²⋊C₄), C₂⁴.₁₁₂(C₂×C₄), C₂².₇C₂²≀C₂, C₂³.₅₄₃(C₂×D₄), C₂.₆(C₂⁴⋊₃C₄), C₂³.₁₈₀(C₂²×C₄), (C₂²×D₄).₄₄₈C₂², (C₂×C₂³⋊C₄)⋊₁C₂, (C₂×C₄)⋊₃(C₂²⋊C₄), C₂.₂₄(C₂×C₂³⋊C₄), (C₂×C₂²⋊C₄)⋊₁C₂², (C₂²×C₄).₇₂(C₂×C₄), C₂².₂₈(C₂×C₂²⋊C₄), SmallGroup(128,513)

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

Derived series

C₁ — C₂³ — C₂⁵⋊C₄

Chief series

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

Lower central

C₁ — C₂ — C₂³ — C₂⁵⋊C₄

Upper central

C₁ — C₂² — C₂⁵ — C₂⁵⋊C₄

Jennings

C₁ — C₂ — C₂⁴ — C₂⁵⋊C₄

Generators and relations for C₂⁵⋊C₄
G = < a,b,c,d,e,f | a²=b²=c²=d²=e²=f⁴=1, ab=ba, ac=ca, ad=da, ae=ea, faf^-1=ace, bc=cb, fbf^-1=bd=db, be=eb, cd=dc, fcf^-1=ce=ec, de=ed, df=fd, ef=fe >

Subgroups: 1108 in 440 conjugacy classes, 72 normal (12 characteristic)
C₁, C₂, C₂, C₂, C₄, C₂², C₂², C₂², C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, C₂³, C₂²⋊C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂⁴, C₂⁴, C₂⁴, C₂³⋊C₄, C₂×C₂²⋊C₄, C₂×C₂²⋊C₄, C₂³×C₄, C₂²×D₄, C₂²×D₄, C₂⁵, C₂⁴⋊₃C₄, C₂×C₂³⋊C₄, D₄×C₂³, C₂⁵⋊C₄
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, C₂³, C₂²⋊C₄, C₂²×C₄, C₂×D₄, C₂³⋊C₄, C₂×C₂²⋊C₄, C₂²≀C₂, C₂⁴⋊₃C₄, C₂×C₂³⋊C₄, C₂⁵⋊C₄

Permutation representations of C₂⁵⋊C₄
►On 16 points - transitive group 16T240

Generators in S₁₆

(1 7)(3 12)(4 14)(5 13)(6 9)(10 15)

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

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

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

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

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

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

G:=TransitiveGroup(16,240);

►On 16 points - transitive group 16T275

Generators in S₁₆

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

(1 8)(4 5)(9 14)(11 16)

(1 5)(4 8)(9 11)(14 16)

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

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

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

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

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

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

G:=TransitiveGroup(16,275);

32 conjugacy classes

class	1	2A	2B	2C	2D	···	2M	2N	···	2S	4A	4B	4C	4D	4E	···	4L
order	1	2	2	2	2	···	2	2	···	2	4	4	4	4	4	···	4
size	1	1	1	1	2	···	2	4	···	4	4	4	4	4	8	···	8

32 irreducible representations

dim	1	1	1	1	1	1	1	2	2	4
type	+	+	+	+				+	+	+
image	C₁	C₂	C₂	C₂	C₄	C₄	C₄	D₄	D₄	C₂³⋊C₄
kernel	C₂⁵⋊C₄	C₂⁴⋊₃C₄	C₂×C₂³⋊C₄	D₄×C₂³	C₂³×C₄	C₂²×D₄	C₂⁵	C₂×D₄	C₂⁴	C₂²
# reps	1	2	4	1	2	4	2	8	4	4

Matrix representation of C₂⁵⋊C₄ ►in GL₆(𝔽₅)

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

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

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

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

C₂⁵⋊C₄ in GAP, Magma, Sage, TeX

C_2^5\rtimes C_4

% in TeX

G:=Group("C2^5:C4");

// GroupNames label

G:=SmallGroup(128,513);

// by ID

G=gap.SmallGroup(128,513);

# by ID

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

// Polycyclic

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

// generators/relations

G = C25⋊C4 order 128 = 27

1st semidirect product of C25 and C4 acting faithfully

G = C₂⁵⋊C₄ order 128 = 2⁷

1^st semidirect product of C₂⁵ and C₄ acting faithfully