C2^5.C2^2 - GroupNames

G = C₂⁵.C₂² order 128 = 2⁷

9^th non-split extension by C₂⁵ of C₂² acting faithfully

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

Aliases: C₂⁴.₂₅D₄, C₂⁵.₉C₂², C₂⁴.₁₇₁C₂³, C₂⁴⋊₇(C₂×C₄), (C₂²×C₄)⋊₁D₄, C₂²≀C₂⋊₄C₄, C₂.₃C₂≀C₂², (C₂×D₄).₇₅D₄, C₂⁴⋊₃C₄⋊₂C₂, C₂².₃₈(C₄×D₄), C₂³⋊₂(C₂²⋊C₄), C₂³.₅₆₄(C₂×D₄), C₂³.₉D₄⋊₆C₂, C₂².₁₁C₂⁴⋊₂C₂, C₂³.₆₇(C₂²×C₄), C₂².₁₀₃C₂²≀C₂, C₂³.₁₁₈(C₄○D₄), C₂².₄₅(C₄⋊D₄), (C₂²×D₄).₂₂C₂², C₂.₂₈(C₂³.₂₃D₄), C₂².₅₂(C₂².D₄), C₂²⋊C₄⋊₅(C₂×C₄), (C₂×C₂³⋊C₄)⋊₄C₂, (C₂×C₄)⋊₂(C₂²⋊C₄), (C₂×D₄).₇₈(C₂×C₄), (C₂×C₂²≀C₂).₁C₂, C₂².₄₄(C₂×C₂²⋊C₄), (C₂×C₂²⋊C₄).₁₀C₂², SmallGroup(128,621)

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

Derived series

C₁ — C₂³ — C₂⁵.C₂²

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂⁴ — C₂²×D₄ — C₂².₁₁C₂⁴ — C₂⁵.C₂²

Lower central

C₁ — C₂ — C₂³ — C₂⁵.C₂²

Upper central

C₁ — C₂² — C₂⁴ — C₂⁵.C₂²

Jennings

C₁ — C₂ — C₂⁴ — C₂⁵.C₂²

Generators and relations for C₂⁵.C₂²
G = < a,b,c,d,e,f,g | a²=b²=c²=d²=e²=g²=1, f²=d, faf^-1=ab=ba, ac=ca, ad=da, ae=ea, gag=ace, bc=cb, bd=db, gbg=be=eb, bf=fb, cd=dc, fcf^-1=ce=ec, cg=gc, gdg=de=ed, df=fd, ef=fe, eg=ge, gfg=cef >

Subgroups: 772 in 289 conjugacy classes, 56 normal (18 characteristic)
C₁, C₂, C₂, C₂, C₄, C₂², C₂², C₂², C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂⁴, C₂⁴, C₂⁴, C₂³⋊C₄, C₂×C₂²⋊C₄, C₂×C₂²⋊C₄, C₂×C₂²⋊C₄, C₄²⋊C₂, C₄×D₄, C₂²≀C₂, C₂²≀C₂, C₂²×D₄, C₂²×D₄, C₂⁵, C₂³.₉D₄, C₂⁴⋊₃C₄, C₂×C₂³⋊C₄, C₂².₁₁C₂⁴, C₂×C₂²≀C₂, C₂⁵.C₂²
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, C₂³, C₂²⋊C₄, C₂²×C₄, C₂×D₄, C₄○D₄, C₂×C₂²⋊C₄, C₄×D₄, C₂²≀C₂, C₄⋊D₄, C₂².D₄, C₂³.₂₃D₄, C₂≀C₂², C₂⁵.C₂²

Permutation representations of C₂⁵.C₂²
►On 16 points - transitive group 16T208

Generators in S₁₆

(6 12)(8 10)

(5 11)(6 12)(7 9)(8 10)

(2 14)(4 16)(5 11)(7 9)

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

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

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

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

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

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

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

G:=TransitiveGroup(16,208);

►On 16 points - transitive group 16T210

Generators in S₁₆

(1 6)(4 8)(9 11)(10 14)(12 16)(13 15)

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

(1 4)(6 8)(10 12)(14 16)

(9 11)(10 12)(13 15)(14 16)

(1 4)(2 3)(5 7)(6 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)

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

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

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

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

G:=TransitiveGroup(16,210);

►On 16 points - transitive group 16T217

Generators in S₁₆

(5 11)(6 8)(7 9)(10 12)

(5 9)(6 10)(7 11)(8 12)

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

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

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

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

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

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

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

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

G:=TransitiveGroup(16,217);

►On 16 points - transitive group 16T247

Generators in S₁₆

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

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

(2 7)(4 5)(10 14)(12 16)

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

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

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

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

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

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

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

G:=TransitiveGroup(16,247);

32 conjugacy classes

class	1	2A	2B	2C	2D	···	2I	2J	···	2O	4A	···	4J	4K	···	4P
order	1	2	2	2	2	···	2	2	···	2	4	···	4	4	···	4
size	1	1	1	1	2	···	2	4	···	4	4	···	4	8	···	8

32 irreducible representations

dim

type

image

C₁

C₂

C₄

D₄

C₄○D₄

C₂≀C₂²

kernel

C₂⁵.C₂²

C₂³.₉D₄

C₂⁴⋊₃C₄

C₂×C₂³⋊C₄

C₂².₁₁C₂⁴

C₂×C₂²≀C₂

C₂²≀C₂

C₂²×C₄

C₂×D₄

C₂⁴

C₂³

C₂

# reps

Matrix representation of C₂⁵.C₂² ►in GL₆(𝔽₅)

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

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

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	0	0	0	4

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

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	0	1	0	0
0	0	1	0	0	0

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

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

C₂⁵.C₂² in GAP, Magma, Sage, TeX

C_2^5.C_2^2

% in TeX

G:=Group("C2^5.C2^2");

// GroupNames label

G:=SmallGroup(128,621);

// by ID

G=gap.SmallGroup(128,621);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,521,2804,1411,1027]);

// Polycyclic

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

// generators/relations

G = C25.C22 order 128 = 27

9th non-split extension by C25 of C22 acting faithfully

G = C₂⁵.C₂² order 128 = 2⁷

9^th non-split extension by C₂⁵ of C₂² acting faithfully