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

G = 2₊¹⁺⁴⋊₂C₄ order 128 = 2⁷

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

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

Aliases: C₂⁴.₁₉D₄, 2₊¹⁺⁴⋊₂C₄, C₂⁴.₁₆₆C₂³, C₂.₁C₂≀C₂², (C₂×D₄).₆₇D₄, (C₂³×C₄)⋊₁C₂², (C₂²×C₄).₅₆D₄, C₂³.₃₀(C₂×D₄), C₂³⋊₁(C₂²⋊C₄), C₂³.₃(C₂²×C₄), C₂².₁₀C₂²≀C₂, C₂³.₂₃D₄⋊₁C₂, (C₂²×D₄).₈C₂², C₂.₁₅(C₂⁴⋊₃C₄), C₂.₁(C₂³.₇D₄), (C₂×2₊¹⁺⁴).₁C₂, (C₂×C₂³⋊C₄)⋊₃C₂, (C₂×C₄)⋊₁(C₂²⋊C₄), (C₂×D₄).₆₂(C₂×C₄), (C₂×C₂²⋊C₄)⋊₂C₂², C₂².₉(C₂×C₂²⋊C₄), SmallGroup(128,522)

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

Derived series

C₁ — C₂³ — 2₊¹⁺⁴⋊₂C₄

Chief series

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

Lower central

C₁ — C₂ — C₂³ — 2₊¹⁺⁴⋊₂C₄

Upper central

C₁ — C₂² — C₂⁴ — 2₊¹⁺⁴⋊₂C₄

Jennings

C₁ — C₂ — C₂⁴ — 2₊¹⁺⁴⋊₂C₄

Generators and relations for 2₊¹⁺⁴⋊₂C₄
G = < a,b,c,d,e | a⁴=b²=d²=e⁴=1, c²=a², bab=a^-1, ac=ca, ad=da, eae^-1=a^-1cd, bc=cb, ede^-1=bd=db, be=eb, dcd=a²c, ece^-1=a²bc >

Subgroups: 780 in 335 conjugacy classes, 68 normal (10 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₂², C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₂³, C₂³, C₂²⋊C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₄○D₄, C₂⁴, C₂⁴, C₂⁴, C₂.C₄², C₂³⋊C₄, C₂×C₂²⋊C₄, C₂×C₂²⋊C₄, C₂³×C₄, C₂²×D₄, C₂²×D₄, C₂×C₄○D₄, 2₊¹⁺⁴, 2₊¹⁺⁴, C₂³.₂₃D₄, C₂×C₂³⋊C₄, C₂×2₊¹⁺⁴, 2₊¹⁺⁴⋊₂C₄
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, C₂³, C₂²⋊C₄, C₂²×C₄, C₂×D₄, C₂×C₂²⋊C₄, C₂²≀C₂, C₂⁴⋊₃C₄, C₂≀C₂², C₂³.₇D₄, 2₊¹⁺⁴⋊₂C₄

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

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

(9 11)(10 12)(13 15)(14 16)(17 28)(18 25)(19 26)(20 27)(21 29)(22 30)(23 31)(24 32)

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

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

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

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

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	1	1	1	1	1	2	2	2	4	4
type	+	+	+	+		+	+	+	+
image	C₁	C₂	C₂	C₂	C₄	D₄	D₄	D₄	C₂≀C₂²	C₂³.₇D₄
kernel	2₊¹⁺⁴⋊₂C₄	C₂³.₂₃D₄	C₂×C₂³⋊C₄	C₂×2₊¹⁺⁴	2₊¹⁺⁴	C₂²×C₄	C₂×D₄	C₂⁴	C₂	C₂
# reps	1	3	3	1	8	3	6	3	2	2

Matrix representation of 2₊¹⁺⁴⋊₂C₄ ►in GL₆(𝔽₅)

4	0	0	0	0	0
3	1	0	0	0	0
0	0	0	4	0	0
0	0	1	0	0	0
0	0	0	0	0	4
0	0	0	0	1	0

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

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

1	4	0	0	0	0
2	4	0	0	0	0
0	0	4	1	4	4
0	0	1	4	4	4
0	0	1	1	1	4
0	0	1	1	4	1

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

2₊¹⁺⁴⋊₂C₄ in GAP, Magma, Sage, TeX

2_+^{1+4}\rtimes_2C_4

% in TeX

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

// GroupNames label

G:=SmallGroup(128,522);

// by ID

G=gap.SmallGroup(128,522);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,422,2019,521,1411,2028]);

// Polycyclic

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

// generators/relations

G = 2+ 1+4⋊2C4 order 128 = 27

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

G = 2₊¹⁺⁴⋊₂C₄ order 128 = 2⁷

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