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G = 2+ 1+4order 32 = 25

Extraspecial group

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

Aliases: 2+ 1+4, D4D4, Q8Q8, D44C22, C2.4C24, C4.9C23, Q84C22, C232C22, C22.2C23, C4○D43C2, (C2×D4)⋊6C2, (C2×C4)⋊2C22, 2-Sylow(Omega+(4,3)), SmallGroup(32,49)

Series: Derived Chief Lower central Upper central Jennings

C1C2 — 2+ 1+4
C1C2C22C23C2×D4 — 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=d2=1, c2=a2, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=a2c >

Subgroups: 110 in 83 conjugacy classes, 68 normal (3 characteristic)
C1, C2, C2 [×9], C4 [×6], C22 [×9], C22 [×6], C2×C4 [×9], D4 [×18], Q8 [×2], C23 [×6], C2×D4 [×9], C4○D4 [×6], 2+ 1+4
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], C24, 2+ 1+4

Character table of 2+ 1+4

 class 12A2B2C2D2E2F2G2H2I2J4A4B4C4D4E4F
 size 11222222222222222
ρ111111111111111111    trivial
ρ2111111-1-1-1-11-1-11-1-11    linear of order 2
ρ3111-1-11-1-1-11-111-11-11    linear of order 2
ρ4111-1-11111-1-1-1-1-1-111    linear of order 2
ρ511-1-1-1-1-111-11111-1-11    linear of order 2
ρ611-111-1-1111-1-1-1-11-11    linear of order 2
ρ711-1-1-1-11-1-111-1-11111    linear of order 2
ρ811-111-11-1-1-1-111-1-111    linear of order 2
ρ9111-11-1-1-11-1-11-1111-1    linear of order 2
ρ101111-1-1-1-1111-11-1-11-1    linear of order 2
ρ11111-11-111-11-1-111-1-1-1    linear of order 2
ρ121111-1-111-1-111-1-11-1-1    linear of order 2
ρ1311-11-11-11-1-1-1-11111-1    linear of order 2
ρ1411-1-111-11-1111-1-1-11-1    linear of order 2
ρ1511-11-111-111-11-11-1-1-1    linear of order 2
ρ1611-1-1111-11-11-11-11-1-1    linear of order 2
ρ174-4000000000000000    orthogonal faithful

Permutation representations of 2+ 1+4
On 8 points - transitive group 8T22
Generators in S8
(1 2 3 4)(5 6 7 8)
(1 4)(2 3)(5 8)(6 7)
(1 5 3 7)(2 6 4 8)
(1 7)(2 8)(3 5)(4 6)

G:=sub<Sym(8)| (1,2,3,4)(5,6,7,8), (1,4)(2,3)(5,8)(6,7), (1,5,3,7)(2,6,4,8), (1,7)(2,8)(3,5)(4,6)>;

G:=Group( (1,2,3,4)(5,6,7,8), (1,4)(2,3)(5,8)(6,7), (1,5,3,7)(2,6,4,8), (1,7)(2,8)(3,5)(4,6) );

G=PermutationGroup([(1,2,3,4),(5,6,7,8)], [(1,4),(2,3),(5,8),(6,7)], [(1,5,3,7),(2,6,4,8)], [(1,7),(2,8),(3,5),(4,6)])

G:=TransitiveGroup(8,22);

On 16 points - transitive group 16T23
Generators in S16
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 14)(2 13)(3 16)(4 15)(5 12)(6 11)(7 10)(8 9)
(1 8 3 6)(2 5 4 7)(9 16 11 14)(10 13 12 15)
(1 6)(2 7)(3 8)(4 5)(9 16)(10 13)(11 14)(12 15)

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

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

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

G:=TransitiveGroup(16,23);

2+ 1+4 is a maximal subgroup of
D4.9D4  C2≀C22  C23.7D4  D4○D8  D4○SD16  C2.C25  Q8.A4  C23⋊A4
 D4⋊D2p: D44D4  D46D6  D4○D12  D46D10  D48D10  D46D14  D48D14  D46D22 ...
2+ 1+4 is a maximal quotient of
C22.11C24  C23.33C23  C233D4  C22.29C24  C22.31C24  C22.32C24  C22.33C24  C22.34C24  C22.36C24  C232Q8  C23.41C23  D42  Q86D4  C22.45C24  C22.47C24  D43Q8  C22.49C24  Q82  C22.53C24  C22.54C24  C24⋊C22  C22.56C24  C22.57C24
 D4⋊D2p: D45D4  D46D6  D4○D12  D46D10  D48D10  D46D14  D48D14  D46D22 ...

Polynomial with Galois group 2+ 1+4 over ℚ
actionf(x)Disc(f)
8T22x8-16x6-12x5+73x4+96x3-56x2-124x-41216·32·54·72·312

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

000-1
0010
0-100
1000
,
1000
0100
00-10
000-1
,
0100
-1000
0001
00-10
,
0100
1000
000-1
00-10
G:=sub<GL(4,Integers())| [0,0,0,1,0,0,-1,0,0,1,0,0,-1,0,0,0],[1,0,0,0,0,1,0,0,0,0,-1,0,0,0,0,-1],[0,-1,0,0,1,0,0,0,0,0,0,-1,0,0,1,0],[0,1,0,0,1,0,0,0,0,0,0,-1,0,0,-1,0] >;

2+ 1+4 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(32,49);
// by ID

G=gap.SmallGroup(32,49);
# by ID

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

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

Export

Character table of 2+ 1+4 in TeX

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