Copied to
clipboard

G = 2- 1+4order 32 = 25

Extraspecial group

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

Aliases: 2- 1+4, D4Q8, C2.5C24, C4.10C23, D4.4C22, Q8.4C22, C22.3C23, C4○D44C2, (C2×Q8)⋊5C2, (C2×C4).8C22, Dirac group of gamma matrices, SmallGroup(32,50)

Series: Derived Chief Lower central Upper central Jennings

C1C2 — 2- 1+4
C1C2C22C2×C4C2×Q8 — 2- 1+4
C1C2 — 2- 1+4
C1C2 — 2- 1+4
C1C2 — 2- 1+4

Generators and relations for 2- 1+4
 G = < a,b,c,d | a4=b2=1, c2=d2=a2, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=a2c >

Subgroups: 78 in 73 conjugacy classes, 68 normal (3 characteristic)
C1, C2, C2 [×5], C4 [×10], C22 [×5], C2×C4 [×15], D4 [×10], Q8 [×10], C2×Q8 [×5], C4○D4 [×10], 2- 1+4
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], C24, 2- 1+4

Character table of 2- 1+4

 class 12A2B2C2D2E2F4A4B4C4D4E4F4G4H4I4J
 size 11222222222222222
ρ111111111111111111    trivial
ρ211-1-11-1-11-11-11-1111-1    linear of order 2
ρ3111-1-1-1-1-111-111-11-11    linear of order 2
ρ411-11-111-1-1111-1-11-1-1    linear of order 2
ρ511-1111-1-11-1-11-1-1-111    linear of order 2
ρ61111-11-11-1-1-1111-1-1-1    linear of order 2
ρ7111-11-11-1-1-1111-1-11-1    linear of order 2
ρ811-1-1-1-1111-111-11-1-11    linear of order 2
ρ911-11-1-1-1-1-111-111-111    linear of order 2
ρ1011111-1-11111-1-1-1-1-1-1    linear of order 2
ρ11111-1-111-111-1-1-11-11-1    linear of order 2
ρ1211-1-11111-11-1-11-1-1-11    linear of order 2
ρ1311-1-1-11-111-11-11-111-1    linear of order 2
ρ14111-111-1-1-1-11-1-111-11    linear of order 2
ρ151111-1-111-1-1-1-1-1-1111    linear of order 2
ρ1611-111-11-11-1-1-1111-1-1    linear of order 2
ρ174-4000000000000000    symplectic faithful, Schur index 2

Permutation representations of 2- 1+4
On 16 points - transitive group 16T20
Generators in S16
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 16)(2 15)(3 14)(4 13)(5 11)(6 10)(7 9)(8 12)
(1 11 3 9)(2 12 4 10)(5 14 7 16)(6 15 8 13)
(1 4 3 2)(5 8 7 6)(9 10 11 12)(13 14 15 16)

G:=sub<Sym(16)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,16)(2,15)(3,14)(4,13)(5,11)(6,10)(7,9)(8,12), (1,11,3,9)(2,12,4,10)(5,14,7,16)(6,15,8,13), (1,4,3,2)(5,8,7,6)(9,10,11,12)(13,14,15,16)>;

G:=Group( (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,16)(2,15)(3,14)(4,13)(5,11)(6,10)(7,9)(8,12), (1,11,3,9)(2,12,4,10)(5,14,7,16)(6,15,8,13), (1,4,3,2)(5,8,7,6)(9,10,11,12)(13,14,15,16) );

G=PermutationGroup([(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(1,16),(2,15),(3,14),(4,13),(5,11),(6,10),(7,9),(8,12)], [(1,11,3,9),(2,12,4,10),(5,14,7,16),(6,15,8,13)], [(1,4,3,2),(5,8,7,6),(9,10,11,12),(13,14,15,16)])

G:=TransitiveGroup(16,20);

Matrix representation of 2- 1+4 in GL4(𝔽3) generated by

1002
0020
0100
2002
,
0120
1001
0001
0010
,
0020
1002
1000
0120
,
0200
1000
2001
0220
G:=sub<GL(4,GF(3))| [1,0,0,2,0,0,1,0,0,2,0,0,2,0,0,2],[0,1,0,0,1,0,0,0,2,0,0,1,0,1,1,0],[0,1,1,0,0,0,0,1,2,0,0,2,0,2,0,0],[0,1,2,0,2,0,0,2,0,0,0,2,0,0,1,0] >;

2- 1+4 in GAP, Magma, Sage, TeX

2_-^{1+4}
% in TeX

G:=Group("ES-(2,2)");
// GroupNames label

G:=SmallGroup(32,50);
// by ID

G=gap.SmallGroup(32,50);
# by ID

G:=PCGroup([5,-2,2,2,2,-2,181,86,157,72,483]);
// Polycyclic

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

׿
×
𝔽