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G = M4(2).37D4  order 128 = 27

1st non-split extension by M4(2) of D4 acting via D4/C22=C2

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

Aliases: M4(2).37D4, M4(2).11C23, C8.25(C2×D4), C4○D4.20D4, D4.15(C2×D4), C8○D4⋊1C22, Q8.15(C2×D4), D4.3D4⋊1C2, D4.4D4⋊4C2, (C2×D4).151D4, (C2×D8)⋊19C22, Q8○M4(2)⋊1C2, (C2×C4).19C24, (C2×C8).22C23, (C2×Q8).127D4, D8⋊C22⋊9C2, C8⋊C22⋊11C22, C8.C4⋊3C22, C4○D4.31C23, (C2×D4).73C23, C4.166(C22×D4), (C2×Q8).61C23, C4.177(C4⋊D4), (C2×SD16)⋊13C22, C4.D4⋊15C22, C8.C22⋊12C22, M4(2).C4⋊13C2, C23.196(C4○D4), C4.10D4⋊15C22, C22.10(C4⋊D4), (C22×C4).287C23, (C22×D4).354C22, (C2×M4(2)).62C22, M4(2).8C22⋊5C2, (C2×C8⋊C22)⋊21C2, (C2×C4).477(C2×D4), C2.90(C2×C4⋊D4), (C2×C4.D4)⋊12C2, C22.22(C2×C4○D4), (C2×C4).479(C4○D4), (C2×C4○D4).132C22, SmallGroup(128,1800)

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

Derived series
C1 — C2×C4 — M4(2).37D4
Chief series
C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C4○D4 — Q8○M4(2) — M4(2).37D4
Lower central
C1 — C2 — C2×C4 — M4(2).37D4
Upper central
C1 — C2 — C22×C4 — M4(2).37D4
Jennings
C1 — C2 — C2 — C2×C4 — M4(2).37D4

Generators and relations for M4(2).37D4
 G = < a,b,c,d | a8=b2=d2=1, c4=a4, bab=a5, cac-1=dad=a-1, bc=cb, bd=db, dcd=a4c3 >

Subgroups: 476 in 233 conjugacy classes, 98 normal (36 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C2×C8, C2×C8, M4(2), M4(2), D8, SD16, Q16, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C4○D4, C4○D4, C24, C4.D4, C4.10D4, C8.C4, C2×M4(2), C2×M4(2), C8○D4, C8○D4, C2×D8, C2×D8, C2×SD16, C2×SD16, C4○D8, C8⋊C22, C8⋊C22, C8.C22, C8.C22, C22×D4, C2×C4○D4, C2×C4.D4, M4(2).8C22, M4(2).C4, D4.3D4, D4.4D4, Q8○M4(2), C2×C8⋊C22, D8⋊C22, M4(2).37D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C4⋊D4, C22×D4, C2×C4○D4, C2×C4⋊D4, M4(2).37D4

Character table of M4(2).37D4

 class 12A2B2C2D2E2F2G2H2I4A4B4C4D4E4F4G8A8B8C8D8E8F8G8H8I8J8K8L
 size 11222448882222448444444448888
ρ111111111111111111111111111111    trivial
ρ211111111-1-11111111-1-1-1-1-1-1-1-1-111-1    linear of order 2
ρ311-11-1-1111-1-111-11-1-1-1-11-111-1111-1-1    linear of order 2
ρ411-11-1-111-11-111-11-1-111-11-1-11-1-11-11    linear of order 2
ρ511111-1-11111111-1-11-11111-1-1-1-1-1-1-1    linear of order 2
ρ611111-1-11-1-11111-1-111-1-1-1-11111-1-11    linear of order 2
ρ711-11-11-111-1-111-1-11-11-11-11-11-1-1-111    linear of order 2
ρ811-11-11-11-11-111-1-11-1-11-11-11-111-11-1    linear of order 2
ρ911-11-11-1-1-11-111-1-1111-11-11-11-111-1-1    linear of order 2
ρ1011-11-11-1-11-1-111-1-111-11-11-11-11-11-11    linear of order 2
ρ1111111-1-1-1-1-11111-1-1-1-11111-1-1-11111    linear of order 2
ρ1211111-1-1-1111111-1-1-11-1-1-1-1111-111-1    linear of order 2
ρ1311-11-1-11-1-11-111-11-11-1-11-111-11-1-111    linear of order 2
ρ1411-11-1-11-11-1-111-11-1111-11-1-11-11-11-1    linear of order 2
ρ151111111-1-1-1111111-111111111-1-1-1-1    linear of order 2
ρ161111111-111111111-1-1-1-1-1-1-1-1-11-1-11    linear of order 2
ρ1722-2-2200000-2-22200002-2-220000000    orthogonal lifted from D4
ρ18222-2-2-2-2000-22-22220000000000000    orthogonal lifted from D4
ρ1922-2-22-2200022-2-2-220000000000000    orthogonal lifted from D4
ρ2022-2-222-200022-2-22-20000000000000    orthogonal lifted from D4
ρ2122-2-2200000-2-2220000-222-20000000    orthogonal lifted from D4
ρ22222-2-222000-22-22-2-20000000000000    orthogonal lifted from D4
ρ23222-2-2000002-22-20000-2-2220000000    orthogonal lifted from D4
ρ24222-2-2000002-22-2000022-2-20000000    orthogonal lifted from D4
ρ2522-22-2000002-2-220002i00002i-2i-2i0000    complex lifted from C4○D4
ρ262222200000-2-2-2-20002i0000-2i-2i2i0000    complex lifted from C4○D4
ρ2722-22-2000002-2-22000-2i0000-2i2i2i0000    complex lifted from C4○D4
ρ282222200000-2-2-2-2000-2i00002i2i-2i0000    complex lifted from C4○D4
ρ298-8000000000000000000000000000    orthogonal faithful

Permutation representations of M4(2).37D4
►On 16 points - transitive group 16T312
Generators in S16
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)

(2 6)(4 8)(10 14)(12 16)

(1 11 7 13 5 15 3 9)(2 10 8 12 6 14 4 16)

(2 8)(3 7)(4 6)(9 11)(12 16)(13 15)
 
G:=sub<Sym(16)| (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (2,6)(4,8)(10,14)(12,16), (1,11,7,13,5,15,3,9)(2,10,8,12,6,14,4,16), (2,8)(3,7)(4,6)(9,11)(12,16)(13,15)>;
 
G:=Group( (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (2,6)(4,8)(10,14)(12,16), (1,11,7,13,5,15,3,9)(2,10,8,12,6,14,4,16), (2,8)(3,7)(4,6)(9,11)(12,16)(13,15) );
 
G=PermutationGroup([[(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(2,6),(4,8),(10,14),(12,16)], [(1,11,7,13,5,15,3,9),(2,10,8,12,6,14,4,16)], [(2,8),(3,7),(4,6),(9,11),(12,16),(13,15)]])
 
G:=TransitiveGroup(16,312);
 

Matrix representation of M4(2).37D4 ►in GL8(ℤ)

0000-1000
00000-100
000000-10
0000000-1
0-1000000
10000000
000-10000
00100000
,
-10000000
0-1000000
00-100000
000-10000
00001000
00000100
00000010
00000001
,
00010000
00100000
10000000
0-1000000
00000010
0000000-1
00000-100
0000-1000
,
10000000
0-1000000
00010000
00100000
00000-100
0000-1000
00000010
0000000-1

G:=sub<GL(8,Integers())| [0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0],[-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0],[1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1] >;
 

M4(2).37D4 in GAP, Magma, Sage, TeX 

M_4(2)._{37}D_4
 
% in TeX
 
G:=Group("M4(2).37D4");
 
// GroupNames label
 
G:=SmallGroup(128,1800);
 
// by ID
 
G=gap.SmallGroup(128,1800);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,1018,2804,172,4037,1027,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^8=b^2=d^2=1,c^4=a^4,b*a*b=a^5,c*a*c^-1=d*a*d=a^-1,b*c=c*b,b*d=d*b,d*c*d=a^4*c^3>;
 
// generators/relations
 

Export

Character table of M4(2).37D4 in TeX

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