C2^5:C2^2

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

Aliases: C₂⁵⋊₂C₂², C₂³.₅₇₉C₂⁴, C₂⁴.₃₈₇C₂³, C₂².₃₅₃2₊⁽¹⁺⁴⁾, (C₂×D₄)⋊₁₆D₄, C₂.₄₃(D₄²), (C₂×C₄²)⋊₃₀C₂², C₂³.₂₀₅(C₂×D₄), C₂³.₁₀D₄⋊₇₆C₂, C₂.₄₀(C₂³⋊₃D₄), (C₂²×C₄).₁₇₇C₂³, C₂².₃₈₈(C₂²×D₄), C₂⁴.₃C₂²⋊₇₄C₂, C₂.C₄²⋊₃₆C₂², C₂.₄(C₂⁴⋊C₂²), (C₂²×D₄).₂₁₈C₂², (C₂×C₄).₈₇(C₂×D₄), (C₂×C₄⋊C₄)⋊₃₂C₂², (C₂×C₂²≀C₂)⋊₁₅C₂, (C₂×C₂²⋊C₄)⋊₂₉C₂², SmallGroup(128,1411)

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

Derived series

C₁ — C₂³ — C₂⁵⋊C₂²

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂⁴ — C₂⁵ — C₂×C₂²≀C₂ — C₂⁵⋊C₂²

Lower central

C₁ — C₂³ — C₂⁵⋊C₂²

Upper central

C₁ — C₂³ — C₂⁵⋊C₂²

Jennings

C₁ — C₂³ — C₂⁵⋊C₂²

Subgroups: 1268 in 500 conjugacy classes, 112 normal (6 characteristic)
C₁, C₂, C₂^[×6], C₂^[×12], C₄^[×12], C₂², C₂²^[×6], C₂²^[×92], C₂×C₄^[×6], C₂×C₄^[×24], D₄^[×36], C₂³, C₂³^[×12], C₂³^[×92], C₄², C₂²⋊C₄^[×36], C₄⋊C₄^[×3], C₂²×C₄^[×9], C₂×D₄^[×12], C₂×D₄^[×36], C₂⁴^[×6], C₂⁴^[×20], C₂.C₄²^[×2], C₂×C₄², C₂×C₂²⋊C₄^[×18], C₂×C₄⋊C₄^[×3], C₂²≀C₂^[×24], C₂²×D₄^[×9], C₂⁵^[×2], C₂⁴.₃C₂²^[×3], C₂³.₁₀D₄^[×6], C₂×C₂²≀C₂^[×6], C₂⁵⋊C₂²

Quotients:
C₁, C₂^[×15], C₂²^[×35], D₄^[×12], C₂³^[×15], C₂×D₄^[×18], C₂⁴, C₂²×D₄^[×3], 2₊⁽¹⁺⁴⁾^[×4], C₂³⋊₃D₄^[×3], D₄²^[×3], C₂⁴⋊C₂², C₂⁵⋊C₂²

Generators and relations
G = < a,b,c,d,e,f,g | a²=b²=c²=d²=e²=f²=g²=1, ab=ba, ac=ca, faf=ad=da, ae=ea, gag=acd, gbg=bc=cb, bd=db, fbf=be=eb, cd=dc, ce=ec, cf=fc, cg=gc, de=ed, df=fd, dg=gd, ef=fe, eg=ge, fg=gf >

Smallest permutation representation
►On 32 points

Generators in S₃₂

(1 2)(3 4)(5 6)(7 8)(9 10)(11 12)(13 14)(15 16)(17 18)(19 20)(21 22)(23 24)(25 26)(27 28)(29 30)(31 32)

(1 15)(2 16)(3 30)(4 29)(5 27)(6 28)(7 17)(8 18)(9 20)(10 19)(11 21)(12 22)(13 31)(14 32)(23 26)(24 25)

(1 31)(2 32)(3 8)(4 7)(5 19)(6 20)(9 28)(10 27)(11 23)(12 24)(13 15)(14 16)(17 29)(18 30)(21 26)(22 25)

(1 27)(2 28)(3 22)(4 21)(5 15)(6 16)(7 26)(8 25)(9 32)(10 31)(11 29)(12 30)(13 19)(14 20)(17 23)(18 24)

(1 30)(2 29)(3 15)(4 16)(5 22)(6 21)(7 14)(8 13)(9 23)(10 24)(11 28)(12 27)(17 32)(18 31)(19 25)(20 26)

(1 30)(2 11)(4 21)(6 16)(7 26)(9 17)(10 24)(12 27)(14 20)(18 31)(23 32)(28 29)

(1 24)(2 29)(3 5)(4 14)(6 26)(7 16)(8 19)(9 23)(10 30)(11 28)(12 31)(13 25)(15 22)(17 32)(18 27)(20 21)

G:=sub<Sym(32)| (1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32), (1,15)(2,16)(3,30)(4,29)(5,27)(6,28)(7,17)(8,18)(9,20)(10,19)(11,21)(12,22)(13,31)(14,32)(23,26)(24,25), (1,31)(2,32)(3,8)(4,7)(5,19)(6,20)(9,28)(10,27)(11,23)(12,24)(13,15)(14,16)(17,29)(18,30)(21,26)(22,25), (1,27)(2,28)(3,22)(4,21)(5,15)(6,16)(7,26)(8,25)(9,32)(10,31)(11,29)(12,30)(13,19)(14,20)(17,23)(18,24), (1,30)(2,29)(3,15)(4,16)(5,22)(6,21)(7,14)(8,13)(9,23)(10,24)(11,28)(12,27)(17,32)(18,31)(19,25)(20,26), (1,30)(2,11)(4,21)(6,16)(7,26)(9,17)(10,24)(12,27)(14,20)(18,31)(23,32)(28,29), (1,24)(2,29)(3,5)(4,14)(6,26)(7,16)(8,19)(9,23)(10,30)(11,28)(12,31)(13,25)(15,22)(17,32)(18,27)(20,21)>;

G:=Group( (1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32), (1,15)(2,16)(3,30)(4,29)(5,27)(6,28)(7,17)(8,18)(9,20)(10,19)(11,21)(12,22)(13,31)(14,32)(23,26)(24,25), (1,31)(2,32)(3,8)(4,7)(5,19)(6,20)(9,28)(10,27)(11,23)(12,24)(13,15)(14,16)(17,29)(18,30)(21,26)(22,25), (1,27)(2,28)(3,22)(4,21)(5,15)(6,16)(7,26)(8,25)(9,32)(10,31)(11,29)(12,30)(13,19)(14,20)(17,23)(18,24), (1,30)(2,29)(3,15)(4,16)(5,22)(6,21)(7,14)(8,13)(9,23)(10,24)(11,28)(12,27)(17,32)(18,31)(19,25)(20,26), (1,30)(2,11)(4,21)(6,16)(7,26)(9,17)(10,24)(12,27)(14,20)(18,31)(23,32)(28,29), (1,24)(2,29)(3,5)(4,14)(6,26)(7,16)(8,19)(9,23)(10,30)(11,28)(12,31)(13,25)(15,22)(17,32)(18,27)(20,21) );

G=PermutationGroup([(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,14),(15,16),(17,18),(19,20),(21,22),(23,24),(25,26),(27,28),(29,30),(31,32)], [(1,15),(2,16),(3,30),(4,29),(5,27),(6,28),(7,17),(8,18),(9,20),(10,19),(11,21),(12,22),(13,31),(14,32),(23,26),(24,25)], [(1,31),(2,32),(3,8),(4,7),(5,19),(6,20),(9,28),(10,27),(11,23),(12,24),(13,15),(14,16),(17,29),(18,30),(21,26),(22,25)], [(1,27),(2,28),(3,22),(4,21),(5,15),(6,16),(7,26),(8,25),(9,32),(10,31),(11,29),(12,30),(13,19),(14,20),(17,23),(18,24)], [(1,30),(2,29),(3,15),(4,16),(5,22),(6,21),(7,14),(8,13),(9,23),(10,24),(11,28),(12,27),(17,32),(18,31),(19,25),(20,26)], [(1,30),(2,11),(4,21),(6,16),(7,26),(9,17),(10,24),(12,27),(14,20),(18,31),(23,32),(28,29)], [(1,24),(2,29),(3,5),(4,14),(6,26),(7,16),(8,19),(9,23),(10,30),(11,28),(12,31),(13,25),(15,22),(17,32),(18,27),(20,21)])

Matrix representation ►G ⊆ GL₆(ℤ)

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	0	-1
0	0	0	0	-1	0

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

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	-1	0
0	0	0	0	0	-1

-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	1	0
0	0	0	0	0	1

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	1	0
0	0	0	0	0	1

-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	1	0
0	0	0	0	0	1

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	1	0
0	0	0	0	0	-1

G:=sub<GL(6,Integers())| [0,1,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-1,0,0,0,0,-1,0],[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],[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,-1,0,0,0,0,0,0,-1],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[-1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1] >;

32 conjugacy classes

class	1	2A	···	2G	2H	···	2S	4A	···	4F	4G	···	4L
order	1	2	···	2	2	···	2	4	···	4	4	···	4
size	1	1	···	1	4	···	4	4	···	4	8	···	8

32 irreducible representations

dim	1	1	1	1	2	4
type	+	+	+	+	+	+
image	C₁	C₂	C₂	C₂	D₄	2₊⁽¹⁺⁴⁾
kernel	C₂⁵⋊C₂²	C₂⁴.₃C₂²	C₂³.₁₀D₄	C₂×C₂²≀C₂	C₂×D₄	C₂²
# reps	1	3	6	6	12	4

In GAP, Magma, Sage, TeX

C_2^5\rtimes C_2^2

% in TeX

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

// GroupNames label

G:=SmallGroup(128,1411);

// by ID

G=gap.SmallGroup(128,1411);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,336,253,758,723,1571,346]);

// Polycyclic

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

// generators/relations

G = C₂⁵⋊C₂² order 128 = 2⁷

2^nd semidirect product of C₂⁵ and C₂² acting faithfully

G = C25⋊C22 order 128 = 27

2nd semidirect product of C25 and C22 acting faithfully

G = C₂⁵⋊C₂² order 128 = 2⁷

2^nd semidirect product of C₂⁵ and C₂² acting faithfully