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G = M4(2)⋊4D4  order 128 = 27

4th semidirect product of M4(2) and D4 acting via D4/C2=C22

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

Aliases: M4(2)⋊4D4, (C2×Q8)⋊8D4, (C2×D4).89D4, C4.64C22≀C2, (C22×C4).76D4, C4.17(C4⋊D4), C23.582(C2×D4), C22.C42⋊8C2, C22.8(C4⋊1D4), C2.25(D4.9D4), C22.200C22≀C2, C22.23(C4⋊D4), C2.12(C23⋊2D4), (C2×C42).346C22, (C22×C4).713C23, C2.25(D4.10D4), (C22×Q8).51C22, C23.67C23⋊6C2, (C2×M4(2)).16C22, C22.31C24.4C2, (C2×C4≀C2)⋊2C2, (C2×C4).37(C2×D4), (C2×C8.C22)⋊2C2, (C2×C4).77(C4○D4), (C2×C4⋊C4).105C22, (C2×C4○D4).48C22, SmallGroup(128,739)

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

Derived series
C1 — C22×C4 — M4(2)⋊4D4
Chief series
C1 — C2 — C22 — C2×C4 — C22×C4 — C2×C4○D4 — C22.31C24 — M4(2)⋊4D4
Lower central
C1 — C2 — C22×C4 — M4(2)⋊4D4
Upper central
C1 — C22 — C22×C4 — M4(2)⋊4D4
Jennings
C1 — C2 — C2 — C22×C4 — M4(2)⋊4D4

Generators and relations for M4(2)⋊4D4
 G = < a,b,c,d | a8=b2=c4=d2=1, bab=a5, cac-1=a5b, dad=a3, bc=cb, dbd=a4b, dcd=c-1 >

Subgroups: 408 in 185 conjugacy classes, 46 normal (18 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), M4(2), SD16, Q16, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C2.C42, C4≀C2, C2×C42, C2×C4⋊C4, C4⋊D4, C22⋊Q8, C2×M4(2), C2×SD16, C2×Q16, C8.C22, C22×Q8, C2×C4○D4, C22.C42, C23.67C23, C2×C4≀C2, C22.31C24, C2×C8.C22, M4(2)⋊4D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C22≀C2, C4⋊D4, C4⋊1D4, C23⋊2D4, D4.9D4, D4.10D4, M4(2)⋊4D4

Character table of M4(2)⋊4D4

 class 12A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J4K4L4M4N8A8B8C8D
 size 11112288222244448888888888
ρ111111111111111111111111111    trivial
ρ2111111-1-11111-1-1-1-1-1-11-1-111111    linear of order 2
ρ31111111-1111111111-1-1-1-1-11-11-1    linear of order 2
ρ4111111-111111-1-1-1-1-11-111-11-11-1    linear of order 2
ρ5111111-1-111111111-1111-11-1-1-1-1    linear of order 2
ρ6111111111111-1-1-1-11-11-111-1-1-1-1    linear of order 2
ρ7111111-1111111111-1-1-1-11-1-11-11    linear of order 2
ρ81111111-11111-1-1-1-111-11-1-1-11-11    linear of order 2
ρ922222200-2-2-2-2000000-20020000    orthogonal lifted from D4
ρ102-2-22-2200-222-20000000000020-2    orthogonal lifted from D4
ρ112222-2-2202-22-20000-2000000000    orthogonal lifted from D4
ρ1222222200-2-2-2-2000000200-20000    orthogonal lifted from D4
ρ132-2-222-200-2-22200000-202000000    orthogonal lifted from D4
ρ142222-2-202-22-2200000000-200000    orthogonal lifted from D4
ρ152222-2-20-2-22-2200000000200000    orthogonal lifted from D4
ρ162-2-22-22002-2-22000000000020-20    orthogonal lifted from D4
ρ172-2-22-2200-222-200000000000-202    orthogonal lifted from D4
ρ182-2-222-200-2-2220000020-2000000    orthogonal lifted from D4
ρ192-2-22-22002-2-220000000000-2020    orthogonal lifted from D4
ρ202222-2-2-202-22-200002000000000    orthogonal lifted from D4
ρ212-2-222-20022-2-22i-2i2i-2i0000000000    complex lifted from C4○D4
ρ222-2-222-20022-2-2-2i2i-2i2i0000000000    complex lifted from C4○D4
ρ234-44-40000000022-2-20000000000    symplectic lifted from D4.10D4, Schur index 2
ρ244-44-400000000-2-2220000000000    symplectic lifted from D4.10D4, Schur index 2
ρ2544-4-4000000002i-2i-2i2i0000000000    complex lifted from D4.9D4
ρ2644-4-400000000-2i2i2i-2i0000000000    complex lifted from D4.9D4

Smallest permutation representation of M4(2)⋊4D4
►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 5)(3 7)(9 13)(11 15)(18 22)(20 24)(25 29)(27 31)

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

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

Matrix representation of M4(2)⋊4D4 ►in GL6(𝔽17)

16160000
010000
0000115
0000116
001000
000100
,
1600000
0160000
0016000
0001600
000010
000001
,
110000
15160000
006200
0071100
00001115
0000106
,
16160000
010000
00001115
0000106
006200
0071100

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,16,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,1,0,0,0,0,15,16,0,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,15,0,0,0,0,1,16,0,0,0,0,0,0,6,7,0,0,0,0,2,11,0,0,0,0,0,0,11,10,0,0,0,0,15,6],[16,0,0,0,0,0,16,1,0,0,0,0,0,0,0,0,6,7,0,0,0,0,2,11,0,0,11,10,0,0,0,0,15,6,0,0] >;
 

M4(2)⋊4D4 in GAP, Magma, Sage, TeX 

M_4(2)\rtimes_4D_4
 
% in TeX
 
G:=Group("M4(2):4D4");
 
// GroupNames label
 
G:=SmallGroup(128,739);
 
// by ID
 
G=gap.SmallGroup(128,739);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,2,-2,448,141,422,387,2019,1018,248,2804,718,172]);
 
// 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^5*b,d*a*d=a^3,b*c=c*b,d*b*d=a^4*b,d*c*d=c^-1>;
 
// generators/relations
 

Export

Character table of M4(2)⋊4D4 in TeX

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