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G = M4(2).5D4order 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(C41D4), C4.C429C2, (C22×SD16)⋊14C2, C2.24(D4.3D4), C23.273(C4○D4), C22.62(C4⋊D4), C2.24(C232D4), (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)⋊C219C2, (C2×C4○D4).54C22, SmallGroup(128,751)

Series: Derived Chief Lower central Upper central Jennings

C1C22×C4 — M4(2).5D4
C1C2C4C2×C4C22×C4C22×D4C2×C8⋊C22 — M4(2).5D4
C1C2C22×C4 — M4(2).5D4
C1C22C22×C4 — M4(2).5D4
C1C2C2C22×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 [×2], C2 [×5], C4 [×4], C4 [×3], C22 [×3], C22 [×13], C8 [×7], C2×C4 [×6], C2×C4 [×7], D4 [×10], Q8 [×8], C23, C23 [×7], C2×C8 [×2], C2×C8 [×5], M4(2) [×4], M4(2) [×4], D8 [×4], SD16 [×16], Q16 [×4], C22×C4, C22×C4 [×2], C2×D4, C2×D4 [×2], C2×D4 [×6], C2×Q8, C2×Q8 [×2], C2×Q8 [×5], C4○D4 [×4], C24, C22⋊C8 [×2], C4.D4 [×2], C4.10D4 [×2], C22×C8, C2×M4(2) [×3], C2×D8, C2×SD16 [×8], C2×Q16, C8⋊C22 [×4], C8.C22 [×4], 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 [×7], C22 [×7], D4 [×12], C23, C2×D4 [×6], C4○D4, C22≀C2 [×3], C4⋊D4 [×3], C41D4, C232D4, D4.3D4 [×2], 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 20)(2 17)(3 22)(4 19)(5 24)(6 21)(7 18)(8 23)(9 25)(10 30)(11 27)(12 32)(13 29)(14 26)(15 31)(16 28)
(1 31 18 9 5 27 22 13)(2 14 19 32 6 10 23 28)(3 25 20 11 7 29 24 15)(4 16 21 26 8 12 17 30)
(1 30 5 26)(2 15 6 11)(3 28 7 32)(4 13 8 9)(10 24 14 20)(12 22 16 18)(17 31 21 27)(19 29 23 25)

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

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

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

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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