ES+(2,2):5C4 - GroupNames

G = 2₊¹⁺⁴⋊₅C₄ order 128 = 2⁷

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

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

Aliases: 2₊¹⁺⁴⋊₅C₄, D₄ ○(D₄⋊C₄), Q₈ ○(Q₈⋊C₄), C₄○D₄.₅₀D₄, C₂.₁(D₄○D₈), C₄.₈(C₂³×C₄), (C₂×D₄).₃₄₃D₄, (C₂×Q₈).₂₆₂D₄, C₄⋊C₄.₃₄₄C₂³, (C₂×C₄).₁₇₈C₂⁴, (C₂×C₈).₄₆₈C₂³, (C₂²×C₈)⋊₄₉C₂², C₂.₁(D₄○SD₁₆), D₄.₂₀(C₂²×C₄), C₂³.₄₃₁(C₂×D₄), C₄.₁₄₃(C₂²×D₄), Q₈.₂₀(C₂²×C₄), D₄⋊C₄⋊₈₆C₂², D₄.₁₉(C₂²⋊C₄), C₄²⋊C₂⋊₄C₂², Q₈⋊C₄⋊₈₉C₂², (C₂×D₄).₃₆₂C₂³, Q₈.₁₉(C₂²⋊C₄), (C₂×Q₈).₃₃₅C₂³, C₂³.₃₇D₄⋊₃₁C₂, C₂³.₃₆D₄⋊₃₆C₂, C₂³.₂₄D₄⋊₃₅C₂, (C₂×M₄(2))⋊₇₁C₂², (C₂²×C₄).₉₀₂C₂³, C₂².₁₂₈(C₂²×D₄), (C₂×2₊¹⁺⁴).₆C₂, C₂³.₃₃C₂³⋊₂C₂, (C₂²×D₄).₃₂₀C₂², C₄○D₄⋊₁(C₂×C₄), (C₂×C₈○D₄)⋊₁₇C₂, (C₂×D₄)⋊₂₄(C₂×C₄), (C₂×C₄⋊C₄)⋊₄₃C₂², (C₂×Q₈)○(Q₈⋊C₄), (C₂×C₄).₄₄₆(C₂×D₄), C₄.₃₃(C₂×C₂²⋊C₄), (C₂×D₄⋊C₄)⋊₅₀C₂, (C₂×C₄).₆₂(C₂²×C₄), C₂².₆(C₂×C₂²⋊C₄), (C₂×C₄○D₄).₈₄C₂², C₂.₄₀(C₂²×C₂²⋊C₄), SmallGroup(128,1629)

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

Derived series

C₁ — C₄ — 2₊¹⁺⁴⋊₅C₄

Chief series

C₁ — C₂ — C₂² — C₂×C₄ — C₂²×C₄ — C₂×C₄○D₄ — C₂×2₊¹⁺⁴ — 2₊¹⁺⁴⋊₅C₄

Lower central

C₁ — C₂ — C₄ — 2₊¹⁺⁴⋊₅C₄

Upper central

C₁ — C₂² — C₂×C₄○D₄ — 2₊¹⁺⁴⋊₅C₄

Jennings

C₁ — C₂ — C₂ — C₂×C₄ — 2₊¹⁺⁴⋊₅C₄

Generators and relations for 2₊¹⁺⁴⋊₅C₄
G = < a,b,c,d,e | a⁴=b²=d²=e⁴=1, c²=a², bab=eae^-1=a^-1, ac=ca, ad=da, bc=cb, bd=db, be=eb, dcd=ece^-1=a²c, ede^-1=cd >

Subgroups: 764 in 389 conjugacy classes, 172 normal (18 characteristic)
C₁, C₂, C₂, C₄, C₄, C₄, C₂², C₂², C₂², C₈, C₂×C₄, C₂×C₄, C₂×C₄, D₄, D₄, Q₈, Q₈, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂×C₈, C₂×C₈, C₂×C₈, M₄(2), C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₄○D₄, C₄○D₄, C₂⁴, D₄⋊C₄, Q₈⋊C₄, Q₈⋊C₄, C₂×C₄⋊C₄, C₄²⋊C₂, C₄×D₄, C₄×Q₈, C₂²×C₈, C₂×M₄(2), C₈○D₄, C₂²×D₄, C₂²×D₄, C₂×C₄○D₄, C₂×C₄○D₄, C₂×C₄○D₄, 2₊¹⁺⁴, 2₊¹⁺⁴, C₂×D₄⋊C₄, C₂³.₂₄D₄, C₂³.₃₆D₄, C₂³.₃₇D₄, C₂³.₃₃C₂³, C₂×C₈○D₄, C₂×2₊¹⁺⁴, 2₊¹⁺⁴⋊₅C₄
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, C₂³, C₂²⋊C₄, C₂²×C₄, C₂×D₄, C₂⁴, C₂×C₂²⋊C₄, C₂³×C₄, C₂²×D₄, C₂²×C₂²⋊C₄, D₄○D₈, D₄○SD₁₆, 2₊¹⁺⁴⋊₅C₄

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

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

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

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

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

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

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

44 conjugacy classes

class	1	2A	2B	2C	2D	···	2I	2J	···	2O	4A	···	4H	4I	···	4R	8A	8B	8C	8D	8E	···	8J
order	1	2	2	2	2	···	2	2	···	2	4	···	4	4	···	4	8	8	8	8	8	···	8
size	1	1	1	1	2	···	2	4	···	4	2	···	2	4	···	4	2	2	2	2	4	···	4

44 irreducible representations

dim

type

image

C₁

C₂

C₄

D₄

D₄○D₈

D₄○SD₁₆

kernel

2₊¹⁺⁴⋊₅C₄

C₂×D₄⋊C₄

C₂³.₂₄D₄

C₂³.₃₆D₄

C₂³.₃₇D₄

C₂³.₃₃C₂³

C₂×C₈○D₄

C₂×2₊¹⁺⁴

2₊¹⁺⁴

C₂×D₄

C₂×Q₈

C₄○D₄

C₂

# reps

Matrix representation of 2₊¹⁺⁴⋊₅C₄ ►in GL₆(𝔽₁₇)

1	0	0	0	0	0
0	1	0	0	0	0
0	0	0	0	0	16
0	0	0	0	1	0
0	0	0	16	0	0
0	0	1	0	0	0

1	0	0	0	0	0
0	1	0	0	0	0
0	0	0	0	0	1
0	0	0	0	16	0
0	0	0	16	0	0
0	0	1	0	0	0

16	0	0	0	0	0
0	16	0	0	0	0
0	0	0	1	0	0
0	0	16	0	0	0
0	0	0	0	0	1
0	0	0	0	16	0

16	0	0	0	0	0
3	1	0	0	0	0
0	0	1	0	0	0
0	0	0	16	0	0
0	0	0	0	16	0
0	0	0	0	0	1

11	13	0	0	0	0
5	6	0	0	0	0
0	0	0	0	3	14
0	0	0	0	14	14
0	0	3	14	0	0
0	0	14	14	0	0

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,1,0,0,0,0,16,0,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16,0,0,0,0,1,0,0,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[16,3,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1],[11,5,0,0,0,0,13,6,0,0,0,0,0,0,0,0,3,14,0,0,0,0,14,14,0,0,3,14,0,0,0,0,14,14,0,0] >;

2₊¹⁺⁴⋊₅C₄ in GAP, Magma, Sage, TeX

2_+^{1+4}\rtimes_5C_4

% in TeX

G:=Group("ES+(2,2):5C4");

// GroupNames label

G:=SmallGroup(128,1629);

// by ID

G=gap.SmallGroup(128,1629);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,521,2804,1411,172]);

// 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,b*d=d*b,b*e=e*b,d*c*d=e*c*e^-1=a^2*c,e*d*e^-1=c*d>;

// generators/relations

G = 2+ 1+4⋊5C4 order 128 = 27

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

G = 2₊¹⁺⁴⋊₅C₄ order 128 = 2⁷

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