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G = M4(2).5D4  order 128 = 27

5th non-split extension by M4(2) of D4 acting via D4/C2=C22

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

Aliases: M4(2).5D4, (C2×D4).97D4, (C2×C8).155D4, (C2×Q8).88D4, C4.14C22≀C2, C4.34(C4⋊1D4), C4.C42⋊9C2, (C22×SD16)⋊14C2, C2.24(D4.3D4), C23.273(C4○D4), C22.62(C4⋊D4), C2.24(C23⋊2D4), (C22×C8).321C22, (C22×C4).717C23, (C22×D4).69C22, (C22×Q8).58C22, (C2×M4(2)).22C22, (C2×C4.D4)⋊4C2, (C2×C4).255(C2×D4), (C2×C8⋊C22).6C2, (C2×C8.C22)⋊4C2, (C2×C4.10D4)⋊3C2, (C22×C8)⋊C2⋊19C2, (C2×C4○D4).54C22, SmallGroup(128,751)

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

Derived series
C1 — C22×C4 — M4(2).5D4
Chief series
C1 — C2 — C4 — C2×C4 — C22×C4 — C22×D4 — C2×C8⋊C22 — M4(2).5D4
Lower central
C1 — C2 — C22×C4 — M4(2).5D4
Upper central
C1 — C22 — C22×C4 — M4(2).5D4
Jennings
C1 — C2 — C2 — C22×C4 — M4(2).5D4

Generators and relations for M4(2).5D4
 G = < a,b,c,d | a8=b2=1, c4=d2=a4, bab=a5, cac-1=a3b, dad-1=ab, cbc-1=a4b, bd=db, dcd-1=c3 >

Subgroups: 424 in 181 conjugacy classes, 46 normal (34 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, 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, C2×Q8, C2×Q8, C4○D4, C24, C22⋊C8, C4.D4, C4.10D4, C22×C8, C2×M4(2), C2×D8, C2×SD16, C2×Q16, C8⋊C22, C8.C22, C22×D4, C22×Q8, C2×C4○D4, C4.C42, (C22×C8)⋊C2, C2×C4.D4, C2×C4.10D4, C22×SD16, C2×C8⋊C22, C2×C8.C22, M4(2).5D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C22≀C2, C4⋊D4, C4⋊1D4, C23⋊2D4, D4.3D4, M4(2).5D4

Character table of M4(2).5D4

 class 12A2B2C2D2E2F2G2H4A4B4C4D4E4F4G8A8B8C8D8E8F8G8H8I8J
 size 11112288822228884444888888
ρ111111111111111111111111111    trivial
ρ2111111-1-1-11111-1-1-111111-111-11    linear of order 2
ρ3111111-1-111111111-1-1-1-11-1-1-1-11    linear of order 2
ρ411111111-11111-1-1-1-1-1-1-111-1-111    linear of order 2
ρ51111111111111-11-1-1-1-1-1-1-111-1-1    linear of order 2
ρ6111111-1-1-111111-11-1-1-1-1-11111-1    linear of order 2
ρ7111111-1-111111-11-11111-11-1-11-1    linear of order 2
ρ811111111-111111-111111-1-1-1-1-1-1    linear of order 2
ρ92-2-222-2000-222-2-2020000000000    orthogonal lifted from D4
ρ102222-2-200222-2-20-200000000000    orthogonal lifted from D4
ρ112-2-22-222-202-22-20000000000000    orthogonal lifted from D4
ρ122222-2-2000-2-22200022-2-2000000    orthogonal lifted from D4
ρ132-2-22-22000-22-22000000020000-2    orthogonal lifted from D4
ρ142-2-22-22-2202-22-20000000000000    orthogonal lifted from D4
ρ152222-2-2000-2-222000-2-222000000    orthogonal lifted from D4
ρ162-2-22-22000-22-220000000-200002    orthogonal lifted from D4
ρ172222-2-200-222-2-20200000000000    orthogonal lifted from D4
ρ182-2-222-20002-2-22000000000-2200    orthogonal lifted from D4
ρ192-2-222-2000-222-220-20000000000    orthogonal lifted from D4
ρ202-2-222-20002-2-220000000002-200    orthogonal lifted from D4
ρ21222222000-2-2-2-2000000002i00-2i0    complex lifted from C4○D4
ρ22222222000-2-2-2-200000000-2i002i0    complex lifted from C4○D4
ρ234-44-40000000000002√-2-2√-200000000    complex lifted from D4.3D4
ρ2444-4-400000000000000-2√-22√-2000000    complex lifted from D4.3D4
ρ2544-4-4000000000000002√-2-2√-2000000    complex lifted from D4.3D4
ρ264-44-4000000000000-2√-22√-200000000    complex lifted from D4.3D4

Smallest permutation representation of M4(2).5D4
►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 24)(2 21)(3 18)(4 23)(5 20)(6 17)(7 22)(8 19)(9 31)(10 28)(11 25)(12 30)(13 27)(14 32)(15 29)(16 26)

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

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

Matrix representation of M4(2).5D4 ►in GL6(𝔽17)

1600000
0160000
0000016
0000160
001000
0001600
,
100000
010000
0016000
0001600
000010
000001
,
400000
13130000
000055
0000512
0051200
00121200
,
480000
13130000
005500
0051200
0000512
00001212

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,16,0,0,0,0,16,0,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[4,13,0,0,0,0,0,13,0,0,0,0,0,0,0,0,5,12,0,0,0,0,12,12,0,0,5,5,0,0,0,0,5,12,0,0],[4,13,0,0,0,0,8,13,0,0,0,0,0,0,5,5,0,0,0,0,5,12,0,0,0,0,0,0,5,12,0,0,0,0,12,12] >;
 

M4(2).5D4 in GAP, Magma, Sage, TeX 

M_4(2)._5D_4
 
% in TeX
 
G:=Group("M4(2).5D4");
 
// GroupNames label
 
G:=SmallGroup(128,751);
 
// by ID
 
G=gap.SmallGroup(128,751);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,2,-2,448,141,422,387,2019,1018,521,248,2804,718,172,4037,2028,1027,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^8=b^2=1,c^4=d^2=a^4,b*a*b=a^5,c*a*c^-1=a^3*b,d*a*d^-1=a*b,c*b*c^-1=a^4*b,b*d=d*b,d*c*d^-1=c^3>;
 
// generators/relations
 

Export

Character table of M4(2).5D4 in TeX

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