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G = M4(2)⋊D4order 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(D44D4), C22.C427C2, C22.7(C41D4), C22.199C22≀C2, C2.27(D4.8D4), C2.11(C232D4), C22.22(C4⋊D4), (C2×C42).345C22, (C22×C4).712C23, C24.3C226C2, (C22×D4).61C22, C22.31C242C2, (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

C1C22×C4 — M4(2)⋊D4
C1C2C22C2×C4C22×C4C2×C4○D4C22.31C24 — M4(2)⋊D4
C1C2C22×C4 — M4(2)⋊D4
C1C22C22×C4 — M4(2)⋊D4
C1C2C2C22×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 [×3], C2 [×6], C4 [×4], C4 [×6], C22 [×3], C22 [×18], C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×14], D4 [×16], Q8 [×4], C23, C23 [×10], C42 [×2], C22⋊C4 [×8], C4⋊C4 [×4], C2×C8 [×2], M4(2) [×4], M4(2) [×2], D8 [×8], SD16 [×8], C22×C4, C22×C4 [×2], C22×C4 [×3], C2×D4 [×4], C2×D4 [×9], C2×Q8 [×2], C4○D4 [×8], C24, C4≀C2 [×4], C2×C42, C2×C22⋊C4 [×2], C2×C4⋊C4, C4⋊D4 [×4], C22⋊Q8 [×2], C2×M4(2) [×2], C2×D8 [×2], C2×SD16 [×2], C8⋊C22 [×8], C22×D4, C2×C4○D4 [×2], C22.C42, C24.3C22, C2×C4≀C2 [×2], C22.31C24, C2×C8⋊C22 [×2], M4(2)⋊D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×12], C23, C2×D4 [×6], C4○D4, C22≀C2 [×3], C4⋊D4 [×3], C41D4, C232D4, D44D4, 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 D44D4
ρ2444-4-400000000002-2-2200000000    orthogonal lifted from D44D4
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)(25 29)(27 31)
(1 14 25 23)(2 15 26 24)(3 12 27 21)(4 13 28 22)(5 10 29 19)(6 11 30 20)(7 16 31 17)(8 9 32 18)
(1 2)(3 8)(4 7)(5 6)(9 21)(10 20)(11 19)(12 18)(13 17)(14 24)(15 23)(16 22)(25 26)(27 32)(28 31)(29 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)(25,29)(27,31), (1,14,25,23)(2,15,26,24)(3,12,27,21)(4,13,28,22)(5,10,29,19)(6,11,30,20)(7,16,31,17)(8,9,32,18), (1,2)(3,8)(4,7)(5,6)(9,21)(10,20)(11,19)(12,18)(13,17)(14,24)(15,23)(16,22)(25,26)(27,32)(28,31)(29,30)>;

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

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

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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