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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, (C2xD4).97D4, (C2xC8).155D4, (C2xQ8).88D4, C4.14C22wrC2, C4.34(C4:1D4), C4.C42:9C2, (C22xSD16):14C2, C2.24(D4.3D4), C23.273(C4oD4), C22.62(C4:D4), C2.24(C23:2D4), (C22xC8).321C22, (C22xC4).717C23, (C22xD4).69C22, (C22xQ8).58C22, (C2xM4(2)).22C22, (C2xC4.D4):4C2, (C2xC4).255(C2xD4), (C2xC8:C22).6C2, (C2xC8.C22):4C2, (C2xC4.10D4):3C2, (C22xC8):C2:19C2, (C2xC4oD4).54C22, SmallGroup(128,751)

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

Derived series
C1 — C22xC4 — M4(2).5D4
Chief series
C1 — C2 — C4 — C2xC4 — C22xC4 — C22xD4 — C2xC8:C22 — M4(2).5D4
Lower central
C1 — C2 — C22xC4 — M4(2).5D4
Upper central
C1 — C22 — C22xC4 — M4(2).5D4
Jennings
C1 — C2 — C2 — C22xC4 — 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, C2xC4, C2xC4, D4, Q8, C23, C23, C2xC8, C2xC8, M4(2), M4(2), D8, SD16, Q16, C22xC4, C22xC4, C2xD4, C2xD4, C2xD4, C2xQ8, C2xQ8, C2xQ8, C4oD4, C24, C22:C8, C4.D4, C4.10D4, C22xC8, C2xM4(2), C2xD8, C2xSD16, C2xQ16, C8:C22, C8.C22, C22xD4, C22xQ8, C2xC4oD4, C4.C42, (C22xC8):C2, C2xC4.D4, C2xC4.10D4, C22xSD16, C2xC8:C22, C2xC8.C22, M4(2).5D4
Quotients: C1, C2, C22, D4, C23, C2xD4, C4oD4, C22wrC2, 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 C4oD4
ρ22222222000-2-2-2-200000000-2i002i0    complex lifted from C4oD4
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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