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G = M4(2).47D4  order 128 = 27

11st non-split extension by M4(2) of D4 acting via D4/C22=C2

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

Aliases: M4(2).47D4, (C2×D8)⋊3C4, C4.2(C4×D4), C4○D4.8D4, (C2×C8).29D4, (C2×SD16)⋊2C4, C4.96C22≀C2, Q8○M4(2)⋊8C2, C22.54(C4×D4), D4.6(C22⋊C4), Q8.6(C22⋊C4), C4.10C42⋊3C2, C4.136(C4⋊D4), M4(2)⋊4C4⋊4C2, (C22×C4).30C23, C42⋊C22⋊13C2, C23.123(C4○D4), C23.37D4⋊23C2, C22.50(C4⋊D4), (C22×D4).29C22, C42⋊C2.28C22, C2.42(C23.23D4), (C2×M4(2)).190C22, C22.6(C22.D4), (C2×C8).6(C2×C4), (C2×D4).85(C2×C4), (C2×C4).239(C2×D4), (C2×C8⋊C22).2C2, C4.19(C2×C22⋊C4), (C2×Q8).73(C2×C4), (C2×C4.D4)⋊18C2, (C2×C4).325(C4○D4), (C2×C4).190(C22×C4), (C2×C4○D4).24C22, SmallGroup(128,635)

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

Derived series
C1 — C2×C4 — M4(2).47D4
Chief series
C1 — C2 — C22 — C2×C4 — C22×C4 — C2×C4○D4 — Q8○M4(2) — M4(2).47D4
Lower central
C1 — C2 — C2×C4 — M4(2).47D4
Upper central
C1 — C2 — C22×C4 — M4(2).47D4
Jennings
C1 — C2 — C2 — C22×C4 — M4(2).47D4

Generators and relations for M4(2).47D4
 G = < a,b,c,d | a8=b2=c4=d2=1, bab=a5, cac-1=ab, dad=a5b, cbc-1=a4b, bd=db, dcd=a2c-1 >

Subgroups: 364 in 162 conjugacy classes, 54 normal (34 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), M4(2), D8, SD16, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C4○D4, C4○D4, C24, C4.D4, D4⋊C4, C4≀C2, C42⋊C2, C2×M4(2), C2×M4(2), C8○D4, C2×D8, C2×SD16, C8⋊C22, C22×D4, C2×C4○D4, C4.10C42, M4(2)⋊4C4, C2×C4.D4, C23.37D4, C42⋊C22, Q8○M4(2), C2×C8⋊C22, M4(2).47D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C4○D4, C2×C22⋊C4, C4×D4, C22≀C2, C4⋊D4, C22.D4, C23.23D4, M4(2).47D4

Character table of M4(2).47D4

 class 12A2B2C2D2E2F2G2H4A4B4C4D4E4F4G4H8A8B8C8D8E8F8G8H8I8J8K8L
 size 11222448822224488444444448888
ρ111111111111111111111111111111    trivial
ρ21111111-1-111111111-1-1-1-1-1-1-1-111-1-1    linear of order 2
ρ311111-1-1111111-1-1-1-1-1-1-1111-1111-1-1    linear of order 2
ρ411111-1-1-1-11111-1-1-1-1111-1-1-11-11111    linear of order 2
ρ51111111-1-1111111-1-111111111-1-1-1-1    linear of order 2
ρ6111111111111111-1-1-1-1-1-1-1-1-1-1-1-111    linear of order 2
ρ711111-1-1-1-11111-1-111-1-1-1111-11-1-111    linear of order 2
ρ811111-1-1111111-1-111111-1-1-11-1-1-1-1-1    linear of order 2
ρ911-11-111-111-11-1-1-1-ii-i-iii-iii-i-ii-11    linear of order 4
ρ1011-11-1-1-1-111-11-111i-iii-ii-ii-i-i-ii1-1    linear of order 4
ρ1111-11-111-111-11-1-1-1i-iii-i-ii-i-iii-i-11    linear of order 4
ρ1211-11-1-1-1-111-11-111-ii-i-ii-ii-iiii-i1-1    linear of order 4
ρ1311-11-1111-11-11-1-1-1-iiii-i-ii-i-ii-ii1-1    linear of order 4
ρ1411-11-1-1-11-11-11-111i-i-i-ii-ii-iii-ii-11    linear of order 4
ρ1511-11-1111-11-11-1-1-1i-i-i-iii-iii-ii-i1-1    linear of order 4
ρ1611-11-1-1-11-11-11-111-iiii-ii-ii-i-ii-i-11    linear of order 4
ρ1722-22-20000-22-220000000-2-22020000    orthogonal lifted from D4
ρ18222-2-2-220022-2-2-2200000000000000    orthogonal lifted from D4
ρ19222-2-20000-2-2220000-222000-200000    orthogonal lifted from D4
ρ2022-2-222-2002-2-22-2200000000000000    orthogonal lifted from D4
ρ2122-2-22-22002-2-222-200000000000000    orthogonal lifted from D4
ρ22222-2-22-20022-2-22-200000000000000    orthogonal lifted from D4
ρ23222-2-20000-2-22200002-2-2000200000    orthogonal lifted from D4
ρ2422-22-20000-22-22000000022-20-20000    orthogonal lifted from D4
ρ2522-2-220000-222-200002i-2i2i000-2i00000    complex lifted from C4○D4
ρ2622-2-220000-222-20000-2i2i-2i0002i00000    complex lifted from C4○D4
ρ27222220000-2-2-2-20000000-2i2i2i0-2i0000    complex lifted from C4○D4
ρ28222220000-2-2-2-200000002i-2i-2i02i0000    complex lifted from C4○D4
ρ298-8000000000000000000000000000    orthogonal faithful

Permutation representations of M4(2).47D4
►On 16 points - transitive group 16T267
Generators in S16
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)

(1 5)(3 7)(10 14)(12 16)

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

(2 6)(3 7)(9 15)(10 16)(11 13)(12 14)
 
G:=sub<Sym(16)| (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,5)(3,7)(10,14)(12,16), (1,11,5,15)(2,12)(3,9,7,13)(4,10)(6,16)(8,14), (2,6)(3,7)(9,15)(10,16)(11,13)(12,14)>;
 
G:=Group( (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,5)(3,7)(10,14)(12,16), (1,11,5,15)(2,12)(3,9,7,13)(4,10)(6,16)(8,14), (2,6)(3,7)(9,15)(10,16)(11,13)(12,14) );
 
G=PermutationGroup([[(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(1,5),(3,7),(10,14),(12,16)], [(1,11,5,15),(2,12),(3,9,7,13),(4,10),(6,16),(8,14)], [(2,6),(3,7),(9,15),(10,16),(11,13),(12,14)]])
 
G:=TransitiveGroup(16,267);
 

Matrix representation of M4(2).47D4 ►in GL8(ℤ)

000000-10
0000000-1
0000-1000
00000-100
00010000
00-100000
01000000
-10000000
,
-10000000
0-1000000
00-100000
000-10000
00001000
00000100
00000010
00000001
,
00001000
00000-100
00000001
00000010
10000000
0-1000000
000-10000
00-100000
,
10000000
0-1000000
000-10000
00-100000
00000-100
0000-1000
00000010
0000000-1

G:=sub<GL(8,Integers())| [0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0],[-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,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0],[1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,-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).47D4 in GAP, Magma, Sage, TeX 

M_4(2)._{47}D_4
 
% in TeX
 
G:=Group("M4(2).47D4");
 
// GroupNames label
 
G:=SmallGroup(128,635);
 
// by ID
 
G=gap.SmallGroup(128,635);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,736,422,2019,521,248,1411,718,172,1027]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^8=b^2=c^4=d^2=1,b*a*b=a^5,c*a*c^-1=a*b,d*a*d=a^5*b,c*b*c^-1=a^4*b,b*d=d*b,d*c*d=a^2*c^-1>;
 
// generators/relations
 

Export

Character table of M4(2).47D4 in TeX

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