ES+(2,2):3C4 - GroupNames

G = 2₊¹⁺⁴⋊₃C₄ order 128 = 2⁷

2^nd semidirect product of 2₊¹⁺⁴ and C₄ acting via C₄/C₂=C₂

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

Aliases: C₂⁴.₂₀D₄, 2₊¹⁺⁴⋊₃C₄, C₄○D₄.₁D₄, C₄.₅₆C₂²≀C₂, D₄⋊₆(C₂²⋊C₄), (C₂×D₄).₂₆₅D₄, Q₈⋊₆(C₂²⋊C₄), (C₂×Q₈).₂₀₈D₄, C₂.₁(D₄⋊₄D₄), (C₂×C₄²)⋊₁₁C₂², C₂³.₅₄₅(C₂×D₄), C₂².₇₅C₂²≀C₂, C₂.₁(D₄.₉D₄), C₂³.₆(C₂²⋊C₄), (C₂²×D₄).₉C₂², C₂.₁₇(C₂⁴⋊₃C₄), C₂³.₃₆D₄⋊₂₉C₂, (C₂×M₄(2))⋊₃₉C₂², (C₂²×C₄).₆₅₇C₂³, C₂⁴.₃C₂²⋊₁C₂, (C₂×2₊¹⁺⁴).₂C₂, (C₂×C₄≀C₂)⋊₁₀C₂, (C₂×C₄⋊C₄)⋊₄C₂², C₄○D₄.₁(C₂×C₄), C₄.₅(C₂×C₂²⋊C₄), (C₂×D₄).₆₄(C₂×C₄), (C₂×C₄).₉₇₅(C₂×D₄), (C₂×C₄).₃(C₂²×C₄), (C₂×C₄.D₄)⋊₁₅C₂, (C₂×C₄○D₄).₅C₂², C₂².₁₁(C₂×C₂²⋊C₄), SmallGroup(128,524)

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

Derived series

C₁ — C₂×C₄ — 2₊¹⁺⁴⋊₃C₄

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂²×D₄ — C₂×2₊¹⁺⁴ — 2₊¹⁺⁴⋊₃C₄

Lower central

C₁ — C₂ — C₂×C₄ — 2₊¹⁺⁴⋊₃C₄

Upper central

C₁ — C₂² — C₂²×C₄ — 2₊¹⁺⁴⋊₃C₄

Jennings

C₁ — C₂ — C₂ — C₂²×C₄ — 2₊¹⁺⁴⋊₃C₄

Generators and relations for 2₊¹⁺⁴⋊₃C₄
G = < a,b,c,d,e | a⁴=b²=d²=e⁴=1, c²=a², bab=eae^-1=a^-1, ac=ca, ad=da, bc=cb, ebe^-1=bd=db, dcd=a²c, ece^-1=acd, de=ed >

Subgroups: 716 in 310 conjugacy classes, 68 normal (18 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₈, C₂×C₄, C₂×C₄, C₂×C₄, D₄, D₄, Q₈, Q₈, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₂×C₈, M₄(2), C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₄○D₄, C₄○D₄, C₂⁴, C₂⁴, C₄.D₄, D₄⋊C₄, Q₈⋊C₄, C₄≀C₂, C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂×M₄(2), C₂²×D₄, C₂²×D₄, C₂×C₄○D₄, C₂×C₄○D₄, 2₊¹⁺⁴, 2₊¹⁺⁴, C₂⁴.₃C₂², C₂×C₄.D₄, C₂³.₃₆D₄, C₂×C₄≀C₂, C₂×2₊¹⁺⁴, 2₊¹⁺⁴⋊₃C₄
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, C₂³, C₂²⋊C₄, C₂²×C₄, C₂×D₄, C₂×C₂²⋊C₄, C₂²≀C₂, C₂⁴⋊₃C₄, D₄⋊₄D₄, D₄.₉D₄, 2₊¹⁺⁴⋊₃C₄

Smallest permutation representation of 2₊¹⁺⁴⋊₃C₄
►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 5)(2 8)(3 7)(4 6)(9 23)(10 22)(11 21)(12 24)(13 14)(15 16)(17 18)(19 20)(25 28)(26 27)(29 30)(31 32)

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

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

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

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

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

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

32 conjugacy classes

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

32 irreducible representations

dim

type

image

C₁

C₂

C₄

D₄

D₄⋊₄D₄

D₄.₉D₄

kernel

2₊¹⁺⁴⋊₃C₄

C₂⁴.₃C₂²

C₂×C₄.D₄

C₂³.₃₆D₄

C₂×C₄≀C₂

C₂×2₊¹⁺⁴

2₊¹⁺⁴

C₂×D₄

C₂×Q₈

C₄○D₄

C₂⁴

C₂

# reps

Matrix representation of 2₊¹⁺⁴⋊₃C₄ ►in GL₆(𝔽₁₇)

16	0	0	0	0	0
0	16	0	0	0	0
0	0	1	1	16	15
0	0	0	0	1	0
0	0	0	16	0	0
0	0	1	1	0	16

1	0	0	0	0	0
0	1	0	0	0	0
0	0	0	0	1	0
0	0	1	1	16	15
0	0	1	0	0	0
0	0	0	0	0	16

16	0	0	0	0	0
15	1	0	0	0	0
0	0	1	1	16	15
0	0	0	0	16	0
0	0	0	1	0	0
0	0	1	0	16	16

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	1	1	16	15
0	0	0	0	0	1

13	4	0	0	0	0
0	4	0	0	0	0
0	0	1	0	0	16
0	0	0	1	0	16
0	0	1	1	16	16
0	0	0	0	0	16

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,1,0,0,1,0,16,1,0,0,16,1,0,0,0,0,15,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,1,0,0,0,0,1,0,0,0,0,1,16,0,0,0,0,0,15,0,16],[16,15,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,1,0,1,0,0,0,16,16,0,16,0,0,15,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,1,0,0,0,1,0,1,0,0,0,0,0,16,0,0,0,0,0,15,1],[13,0,0,0,0,0,4,4,0,0,0,0,0,0,1,0,1,0,0,0,0,1,1,0,0,0,0,0,16,0,0,0,16,16,16,16] >;

2₊¹⁺⁴⋊₃C₄ in GAP, Magma, Sage, TeX

2_+^{1+4}\rtimes_3C_4

% in TeX

G:=Group("ES+(2,2):3C4");

// GroupNames label

G:=SmallGroup(128,524);

// by ID

G=gap.SmallGroup(128,524);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,422,2019,1018,521,248,1411]);

// Polycyclic

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

// generators/relations

G = 2+ 1+4⋊3C4 order 128 = 27

2nd semidirect product of 2+ 1+4 and C4 acting via C4/C2=C2

G = 2₊¹⁺⁴⋊₃C₄ order 128 = 2⁷

2^nd semidirect product of 2₊¹⁺⁴ and C₄ acting via C₄/C₂=C₂