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G = M4(2)⋊5D4order 128 = 27

5th semidirect product of M4(2) and D4 acting via D4/C2=C22

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

Aliases: M4(2)⋊5D4, (C2×C8)⋊1D4, (C2×D4)⋊8D4, (C2×Q8)⋊9D4, C4.5C22≀C2, C22⋊C4.3D4, C4.18(C4⋊D4), D8⋊C221C2, C23.137(C2×D4), M4(2)⋊4C48C2, C42⋊C221C2, C22.29C242C2, C22.37C22≀C2, C23.37D41C2, C22.9(C41D4), (C22×C4).39C23, C2.13(C232D4), C22.59(C4⋊D4), C23.C239C2, (C22×D4).62C22, C42⋊C2.50C22, (C2×M4(2)).17C22, (C2×C8⋊C22)⋊3C2, (C2×C4).251(C2×D4), (C2×C4).337(C4○D4), (C2×C4○D4).49C22, SmallGroup(128,740)

Series: Derived Chief Lower central Upper central Jennings

C1C22×C4 — M4(2)⋊5D4
C1C2C22C2×C4C22×C4C22×D4C22.29C24 — M4(2)⋊5D4
C1C2C22×C4 — M4(2)⋊5D4
C1C2C22×C4 — M4(2)⋊5D4
C1C2C2C22×C4 — M4(2)⋊5D4

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

Subgroups: 488 in 190 conjugacy classes, 44 normal (36 characteristic)
C1, C2, C2 [×7], C4 [×4], C4 [×5], C22 [×3], C22 [×16], C8 [×4], C2×C4 [×6], C2×C4 [×9], D4 [×18], Q8 [×4], C23, C23 [×8], C42 [×2], C22⋊C4 [×2], C22⋊C4 [×5], C4⋊C4 [×2], C2×C8 [×2], C2×C8, M4(2) [×2], M4(2) [×3], D8 [×6], SD16 [×8], Q16 [×2], C22×C4, C22×C4 [×2], C2×D4 [×2], C2×D4 [×2], C2×D4 [×11], C2×Q8 [×2], C4○D4 [×8], C24, C23⋊C4 [×2], D4⋊C4 [×2], C4≀C2 [×2], C42⋊C2 [×2], C22≀C2 [×2], C4⋊D4 [×2], C4.4D4, C41D4, C2×M4(2) [×2], C2×D8, C2×SD16, C4○D8 [×2], C8⋊C22 [×6], C8.C22 [×2], C22×D4, C2×C4○D4 [×2], M4(2)⋊4C4, C23.C23, C23.37D4, C42⋊C22, C22.29C24, C2×C8⋊C22, D8⋊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, M4(2)⋊5D4

Character table of M4(2)⋊5D4

 class 12A2B2C2D2E2F2G2H4A4B4C4D4E4F4G4H4I4J8A8B8C8D
 size 11222888822228888888888
ρ111111111111111111111111    trivial
ρ2111111-1-1-11111-11-1-1111-11-1    linear of order 2
ρ311111-1-1-1-111111-11-1-1-11111    linear of order 2
ρ411111-11111111-1-1-11-1-11-11-1    linear of order 2
ρ51111111-1-11111-11-11-1-1-11-11    linear of order 2
ρ6111111-1111111111-1-1-1-1-1-1-1    linear of order 2
ρ711111-1-1111111-1-1-1-111-11-11    linear of order 2
ρ811111-11-1-111111-11111-1-1-1-1    linear of order 2
ρ922-22-22000-2-2220-200000000    orthogonal lifted from D4
ρ1022-22-2002-222-2-20000000000    orthogonal lifted from D4
ρ11222220000-2-2-2-220-20000000    orthogonal lifted from D4
ρ1222-2-220-2002-22-20002000000    orthogonal lifted from D4
ρ13222-2-200002-2-2200000020-20    orthogonal lifted from D4
ρ1422-22-200-2222-2-20000000000    orthogonal lifted from D4
ρ1522-22-2-2000-2-2220200000000    orthogonal lifted from D4
ρ1622-2-2202002-22-2000-2000000    orthogonal lifted from D4
ρ17222-2-20000-222-20000000-202    orthogonal lifted from D4
ρ18222-2-200002-2-22000000-2020    orthogonal lifted from D4
ρ19222220000-2-2-2-2-2020000000    orthogonal lifted from D4
ρ20222-2-20000-222-2000000020-2    orthogonal lifted from D4
ρ2122-2-220000-22-2200002i-2i0000    complex lifted from C4○D4
ρ2222-2-220000-22-220000-2i2i0000    complex lifted from C4○D4
ρ238-8000000000000000000000    orthogonal faithful

Permutation representations of M4(2)⋊5D4
On 16 points - transitive group 16T331
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 15)(2 16 6 12)(3 13)(4 14 8 10)(5 11)(7 9)
(1 15)(2 10)(3 9)(4 12)(5 11)(6 14)(7 13)(8 16)

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

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

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

G:=TransitiveGroup(16,331);

On 16 points - transitive group 16T367
Generators in S16
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(1 13)(2 10)(3 15)(4 12)(5 9)(6 14)(7 11)(8 16)
(2 14 6 10)(3 7)(4 12 8 16)(9 13)
(1 11)(3 13)(5 15)(7 9)(10 14)(12 16)

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

G:=Group( (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,13)(2,10)(3,15)(4,12)(5,9)(6,14)(7,11)(8,16), (2,14,6,10)(3,7)(4,12,8,16)(9,13), (1,11)(3,13)(5,15)(7,9)(10,14)(12,16) );

G=PermutationGroup([(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(1,13),(2,10),(3,15),(4,12),(5,9),(6,14),(7,11),(8,16)], [(2,14,6,10),(3,7),(4,12,8,16),(9,13)], [(1,11),(3,13),(5,15),(7,9),(10,14),(12,16)])

G:=TransitiveGroup(16,367);

Matrix representation of M4(2)⋊5D4 in GL8(ℤ)

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

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

M4(2)⋊5D4 in GAP, Magma, Sage, TeX

M_4(2)\rtimes_5D_4
% in TeX

G:=Group("M4(2):5D4");
// GroupNames label

G:=SmallGroup(128,740);
// by ID

G=gap.SmallGroup(128,740);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,141,422,387,2019,1018,521,248,2804,718,172,4037]);
// 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^-1*b,c*b*c^-1=d*b*d=a^4*b,d*c*d=c^-1>;
// generators/relations

Export

Character table of M4(2)⋊5D4 in TeX

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