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G = M4(2)⋊D4  order 128 = 27

3rd semidirect product of M4(2) and D4 acting via D4/C2=C22

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

Aliases: M4(2)⋊3D4, (C2×D4)⋊7D4, (C2×Q8).80D4, C4.63C22≀C2, (C22×C4).75D4, C4.16(C4⋊D4), C23.581(C2×D4), C2.31(D4⋊4D4), C22.C42⋊7C2, C22.7(C4⋊1D4), C22.199C22≀C2, C2.27(D4.8D4), C2.11(C23⋊2D4), C22.22(C4⋊D4), (C2×C42).345C22, (C22×C4).712C23, C24.3C22⋊6C2, (C22×D4).61C22, C22.31C24⋊2C2, (C2×M4(2)).15C22, (C2×C4≀C2)⋊1C2, (C2×C8⋊C22)⋊2C2, (C2×C4).36(C2×D4), (C2×C4).76(C4○D4), (C2×C4⋊C4).104C22, (C2×C4○D4).47C22, SmallGroup(128,738)

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

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

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

Subgroups: 504 in 203 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), D8, SD16, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, C4≀C2, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C4⋊D4, C22⋊Q8, C2×M4(2), C2×D8, C2×SD16, C8⋊C22, C22×D4, C2×C4○D4, C22.C42, C24.3C22, C2×C4≀C2, C22.31C24, C2×C8⋊C22, M4(2)⋊D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C22≀C2, C4⋊D4, C4⋊1D4, C23⋊2D4, D4⋊4D4, D4.8D4, M4(2)⋊D4

Character table of M4(2)⋊D4

 class 12A2B2C2D2E2F2G2H2I4A4B4C4D4E4F4G4H4I4J4K4L8A8B8C8D
 size 11112288882222444488888888
ρ111111111111111111111111111    trivial
ρ2111111-1-11-1111111111-1-1-11-11-1    linear of order 2
ρ31111111-1111111-1-1-1-11-1-1-1-11-11    linear of order 2
ρ4111111-111-11111-1-1-1-11111-1-1-1-1    linear of order 2
ρ511111111-111111-1-1-1-1-1-1-111-11-1    linear of order 2
ρ6111111-1-1-1-11111-1-1-1-1-111-11111    linear of order 2
ρ71111111-1-1111111111-111-1-1-1-1-1    linear of order 2
ρ8111111-11-1-111111111-1-1-11-11-11    linear of order 2
ρ92222220000-2-2-2-2000002-200000    orthogonal lifted from D4
ρ102222-2-20200-2-2220000000-20000    orthogonal lifted from D4
ρ112222-2-20-200-2-222000000020000    orthogonal lifted from D4
ρ122-2-22-2200002-2-2200000000020-2    orthogonal lifted from D4
ρ132-2-22-220000-222-20000000020-20    orthogonal lifted from D4
ρ142-2-222-2200-22-22-2000000000000    orthogonal lifted from D4
ρ152222220000-2-2-2-200000-2200000    orthogonal lifted from D4
ρ162-2-22-2200002-2-22000000000-202    orthogonal lifted from D4
ρ172222-2-200-2022-2-2000020000000    orthogonal lifted from D4
ρ182-2-222-2-20022-22-2000000000000    orthogonal lifted from D4
ρ192222-2-2002022-2-20000-20000000    orthogonal lifted from D4
ρ202-2-22-220000-222-200000000-2020    orthogonal lifted from D4
ρ212-2-222-20000-22-222i2i-2i-2i00000000    complex lifted from C4○D4
ρ222-2-222-20000-22-22-2i-2i2i2i00000000    complex lifted from C4○D4
ρ2344-4-40000000000-222-200000000    orthogonal lifted from D4⋊4D4
ρ2444-4-400000000002-2-2200000000    orthogonal lifted from D4⋊4D4
ρ254-44-40000000000-2i2i-2i2i00000000    complex lifted from D4.8D4
ρ264-44-400000000002i-2i2i-2i00000000    complex lifted from D4.8D4

Smallest permutation representation of M4(2)⋊D4
►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)(10 14)(12 16)(17 21)(19 23)(26 30)(28 32)

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

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

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

400000
0130000
0000130
0000013
000400
0013000
,
1600000
0160000
0016000
0001600
000010
000001
,
040000
400000
0001300
0013000
000004
000040
,
0130000
400000
000004
000040
0001300
0013000

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

M4(2)⋊D4 in GAP, Magma, Sage, TeX 

M_4(2)\rtimes D_4
 
% in TeX
 
G:=Group("M4(2):D4");
 
// GroupNames label
 
G:=SmallGroup(128,738);
 
// by ID
 
G=gap.SmallGroup(128,738);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,2,-2,141,422,387,2019,1018,248,2804,718,172]);
 
// 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*b,d*a*d=a^-1,b*c=c*b,d*b*d=a^4*b,d*c*d=c^-1>;
 
// generators/relations
 

Export

Character table of M4(2)⋊D4 in TeX

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