C4:ES+(2,2)

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

Aliases: C₄⋊₁2₊⁽¹⁺⁴⁾, C₂².₈₅C₂⁵, C₂³.₄₁C₂⁴, C₄².₅₇₃C₂³, C₂⁴.₁₃₄C₂³, (D₄²)⋊₁₃C₂, D₄⋊₁₅(C₂×D₄), (C₂×D₄)⋊₃₉D₄, C₂³⋊₃(C₂×D₄), D₄ ○₂(C₄⋊D₄), C₄⋊Q₈⋊₉₀C₂², D₄⋊₆D₄⋊₂₁C₂, D₄⋊₅D₄⋊₁₈C₂, (C₄×D₄)⋊₄₃C₂², C₂³⋊₃D₄⋊₇C₂, (C₂×C₄).₇₆C₂⁴, C₂.₃₁(D₄×C₂³), C₂²≀C₂⋊₇C₂², C₄⋊₁D₄⋊₅₀C₂², C₄⋊D₄⋊₂₆C₂², C₄⋊C₄.₂₉₃C₂³, (C₂³×C₄)⋊₄₂C₂², (C₂×C₄²)⋊₅₈C₂², C₄.₁₂₀(C₂²×D₄), C₂²⋊Q₈⋊₃₁C₂², (C₂×D₄).₄₆₉C₂³, C₄.₄D₄⋊₈₁C₂², (C₂²×D₄)⋊₃₆C₂², C₂²⋊C₄.₂₀C₂³, (C₂×Q₈).₄₄₅C₂³, C₂².₁₃(C₂²×D₄), (C₂²×C₄).₃₅₈C₂³, (C₂×2₊⁽¹⁺⁴⁾)⋊₁₀C₂, C₂².D₄⋊₆C₂², C₂.₃₁(C₂×2₊⁽¹⁺⁴⁾), C₂.₂₂(C₂.C₂⁵), C₂².₂₆C₂⁴⋊₃₄C₂, C₂².₃₁C₂⁴⋊₁₅C₂, (C₂×C₄×D₄)⋊₉₀C₂, (C₂×C₄)⋊₄(C₂×D₄), (C₂×C₄⋊D₄)⋊₆₇C₂, (C₂×C₄⋊C₄)⋊₇₃C₂², (C₂×C₄○D₄)⋊₂₉C₂², (C₂×C₂²⋊C₄)⋊₄₈C₂², SmallGroup(128,2228)

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

Derived series

C₁ — C₂² — C₄⋊2₊⁽¹⁺⁴⁾

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂⁴ — C₂³×C₄ — C₂×C₄×D₄ — C₄⋊2₊⁽¹⁺⁴⁾

Lower central

C₁ — C₂² — C₄⋊2₊⁽¹⁺⁴⁾

Upper central

C₁ — C₂² — C₄⋊2₊⁽¹⁺⁴⁾

Jennings

C₁ — C₂² — C₄⋊2₊⁽¹⁺⁴⁾

Subgroups: 1460 in 830 conjugacy classes, 430 normal (16 characteristic)
C₁, C₂^[×3], C₂^[×18], C₄^[×6], C₄^[×15], C₂², C₂²^[×10], C₂²^[×58], C₂×C₄^[×2], C₂×C₄^[×20], C₂×C₄^[×49], D₄^[×16], D₄^[×76], Q₈^[×8], C₂³, C₂³^[×20], C₂³^[×36], C₄²^[×4], C₂²⋊C₄^[×40], C₄⋊C₄^[×20], C₂²×C₄^[×3], C₂²×C₄^[×26], C₂²×C₄^[×4], C₂×D₄^[×56], C₂×D₄^[×72], C₂×Q₈^[×4], C₄○D₄^[×48], C₂⁴^[×10], C₂×C₄², C₂×C₂²⋊C₄^[×10], C₂×C₄⋊C₄, C₂×C₄⋊C₄^[×4], C₄×D₄^[×16], C₂²≀C₂^[×16], C₄⋊D₄^[×40], C₂²⋊Q₈^[×8], C₂².D₄^[×16], C₄.₄D₄^[×4], C₄⋊₁D₄^[×2], C₄⋊Q₈^[×2], C₂³×C₄^[×2], C₂²×D₄, C₂²×D₄^[×20], C₂×C₄○D₄^[×12], 2₊⁽¹⁺⁴⁾^[×16], C₂×C₄×D₄, C₂×C₄⋊D₄^[×4], C₂².₂₆C₂⁴^[×2], C₂³⋊₃D₄^[×4], C₂².₃₁C₂⁴^[×2], D₄²^[×4], D₄⋊₅D₄^[×8], D₄⋊₆D₄^[×4], C₂×2₊⁽¹⁺⁴⁾^[×2], C₄⋊2₊⁽¹⁺⁴⁾

Quotients:
C₁, C₂^[×31], C₂²^[×155], D₄^[×8], C₂³^[×155], C₂×D₄^[×28], C₂⁴^[×31], C₂²×D₄^[×14], 2₊⁽¹⁺⁴⁾^[×2], C₂⁵, D₄×C₂³, C₂×2₊⁽¹⁺⁴⁾, C₂.C₂⁵, C₄⋊2₊⁽¹⁺⁴⁾

Generators and relations
G = < a,b,c,d,e | a⁴=b⁴=c²=e²=1, d²=b², bab^-1=a^-1, ac=ca, ad=da, ae=ea, cbc=b^-1, bd=db, be=eb, cd=dc, ce=ec, ede=b²d >

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

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

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

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

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

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

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

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	1	0
0	0	0	0	0	1

0	1	0	0	0	0
1	0	0	0	0	0
0	0	-1	2	0	0
0	0	-1	1	0	0
0	0	0	1	0	1
0	0	1	-1	-1	0

-1	0	0	0	0	0
0	-1	0	0	0	0
0	0	-1	2	0	0
0	0	0	1	0	0
0	0	0	1	0	1
0	0	0	-1	1	0

1	0	0	0	0	0
0	1	0	0	0	0
0	0	-1	0	2	0
0	0	0	0	1	1
0	0	-1	0	1	0
0	0	1	-1	-1	0

-1	0	0	0	0	0
0	-1	0	0	0	0
0	0	-1	0	2	0
0	0	0	0	1	1
0	0	0	0	1	0
0	0	0	1	-1	0

G:=sub<GL(6,Integers())| [0,-1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,-1,-1,0,1,0,0,2,1,1,-1,0,0,0,0,0,-1,0,0,0,0,1,0],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,2,1,1,-1,0,0,0,0,0,1,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,-1,1,0,0,0,0,0,-1,0,0,2,1,1,-1,0,0,0,1,0,0],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,2,1,1,-1,0,0,0,1,0,0] >;

44 conjugacy classes

class	1	2A	2B	2C	2D	···	2M	2N	···	2U	4A	···	4H	4I	···	4V
order	1	2	2	2	2	···	2	2	···	2	4	···	4	4	···	4
size	1	1	1	1	2	···	2	4	···	4	2	···	2	4	···	4

44 irreducible representations

dim

type

image

C₁

C₂

D₄

2₊⁽¹⁺⁴⁾

C₂.C₂⁵

kernel

C₄⋊2₊⁽¹⁺⁴⁾

C₂×C₄×D₄

C₂×C₄⋊D₄

C₂².₂₆C₂⁴

C₂³⋊₃D₄

C₂².₃₁C₂⁴

D₄²

D₄⋊₅D₄

D₄⋊₆D₄

C₂×2₊⁽¹⁺⁴⁾

C₂×D₄

C₄

C₂

# reps

In GAP, Magma, Sage, TeX

C_4\rtimes 2_+^{(1+4)}

% in TeX

G:=Group("C4:ES+(2,2)");

// GroupNames label

G:=SmallGroup(128,2228);

// by ID

G=gap.SmallGroup(128,2228);

# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,477,1430,352,570,1684]);

// Polycyclic

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

// generators/relations

G = C₄⋊2₊⁽¹⁺⁴⁾ order 128 = 2⁷

The semidirect product of C₄ and 2₊⁽¹⁺⁴⁾ acting via 2₊⁽¹⁺⁴⁾/C₂×D₄=C₂

G = C4⋊2+ (1+4) order 128 = 27

The semidirect product of C4 and 2+ (1+4) acting via 2+ (1+4)/C2×D4=C2

G = C₄⋊2₊⁽¹⁺⁴⁾ order 128 = 2⁷

The semidirect product of C₄ and 2₊⁽¹⁺⁴⁾ acting via 2₊⁽¹⁺⁴⁾/C₂×D₄=C₂