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G = M4(2)⋊6D4order 128 = 27

6th semidirect product of M4(2) and D4 acting via D4/C2=C22

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

Aliases: M4(2)⋊6D4, C24.32D4, C4⋊C4.92D4, (C2×Q8).92D4, (C2×D4).101D4, C426C432C2, C4.20(C4⋊D4), C23.589(C2×D4), C2.33(D44D4), M4(2)⋊C45C2, C4.40(C4.4D4), C2.27(D4.9D4), C22.219C22≀C2, C23.36D432C2, C22.66(C4⋊D4), (C2×C42).360C22, (C22×C4).722C23, C24.3C229C2, C22.29C24.5C2, (C22×D4).78C22, C42⋊C2.56C22, C4.69(C22.D4), C2.10(C23.10D4), (C2×M4(2)).223C22, C22.32(C22.D4), (C2×C4≀C2)⋊3C2, (C2×C4.D4)⋊5C2, (C2×C4).257(C2×D4), (C2×C4).78(C4○D4), (C2×C4⋊C4).122C22, (C2×C4○D4).57C22, SmallGroup(128,769)

Series: Derived Chief Lower central Upper central Jennings

C1C22×C4 — M4(2)⋊6D4
C1C2C22C2×C4C22×C4C2×C4⋊C4C24.3C22 — M4(2)⋊6D4
C1C2C22×C4 — M4(2)⋊6D4
C1C22C22×C4 — M4(2)⋊6D4
C1C2C2C22×C4 — M4(2)⋊6D4

Generators and relations for M4(2)⋊6D4
 G = < a,b,c,d | a8=b2=c4=d2=1, bab=a5, cac-1=a-1, dad=a3b, cbc-1=dbd=a4b, dcd=c-1 >

Subgroups: 424 in 163 conjugacy classes, 42 normal (38 characteristic)
C1, C2 [×3], C2 [×5], C4 [×4], C4 [×6], C22 [×3], C22 [×17], C8 [×3], C2×C4 [×6], C2×C4 [×12], D4 [×12], Q8 [×2], C23, C23 [×11], C42 [×3], C22⋊C4 [×9], C4⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×2], M4(2) [×2], M4(2) [×3], C22×C4, C22×C4 [×3], C2×D4, C2×D4 [×11], C2×Q8, C4○D4 [×4], C24 [×2], C4.D4 [×2], D4⋊C4, Q8⋊C4, C4≀C2 [×2], C4.Q8, C2.D8, C2×C42, C2×C22⋊C4 [×2], C2×C4⋊C4, C42⋊C2, C22≀C2 [×2], C4⋊D4 [×2], C4.4D4, C41D4, C2×M4(2) [×2], C22×D4, C2×C4○D4, C426C4, C24.3C22, C2×C4.D4, C23.36D4, C2×C4≀C2, M4(2)⋊C4, C22.29C24, M4(2)⋊6D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×8], C23, C2×D4 [×4], C4○D4 [×3], C22≀C2, C4⋊D4 [×3], C22.D4 [×2], C4.4D4, C23.10D4, D44D4, D4.9D4, M4(2)⋊6D4

Character table of M4(2)⋊6D4

 class 12A2B2C2D2E2F2G2H4A4B4C4D4E4F4G4H4I4J4K4L4M8A8B8C8D
 size 11112288822224444888888888
ρ111111111111111111111111111    trivial
ρ2111111-111111111111-1-1-11-1-1-1-1    linear of order 2
ρ3111111-1-1-111111111-1-111-11-1-11    linear of order 2
ρ41111111-1-111111111-11-1-1-1-111-1    linear of order 2
ρ5111111-1-1-11111-1-1-1-11-1111-111-1    linear of order 2
ρ61111111-1-11111-1-1-1-111-1-111-1-11    linear of order 2
ρ71111111111111-1-1-1-1-1111-1-1-1-1-1    linear of order 2
ρ8111111-1111111-1-1-1-1-1-1-1-1-11111    linear of order 2
ρ92-2-222-200022-2-200000000002-20    orthogonal lifted from D4
ρ102222-2-2000-22-22000000-2200000    orthogonal lifted from D4
ρ112222220-22-2-2-2-20000000000000    orthogonal lifted from D4
ρ122222-2-2000-22-220000002-200000    orthogonal lifted from D4
ρ132222-2-2-2002-22-20000020000000    orthogonal lifted from D4
ρ142222-2-22002-22-200000-20000000    orthogonal lifted from D4
ρ1522222202-2-2-2-2-20000000000000    orthogonal lifted from D4
ρ162-2-222-200022-2-20000000000-220    orthogonal lifted from D4
ρ172-2-22-22000-222-20000-2i0002i0000    complex lifted from C4○D4
ρ182-2-22-220002-2-22-2i2i-2i2i000000000    complex lifted from C4○D4
ρ192-2-22-22000-222-200002i000-2i0000    complex lifted from C4○D4
ρ202-2-222-2000-2-222000000000-2i002i    complex lifted from C4○D4
ρ212-2-22-220002-2-222i-2i2i-2i000000000    complex lifted from C4○D4
ρ222-2-222-2000-2-2220000000002i00-2i    complex lifted from C4○D4
ρ234-44-400000000022-2-2000000000    orthogonal lifted from D44D4
ρ244-44-4000000000-2-222000000000    orthogonal lifted from D44D4
ρ2544-4-4000000000-2i2i2i-2i000000000    complex lifted from D4.9D4
ρ2644-4-40000000002i-2i-2i2i000000000    complex lifted from D4.9D4

Smallest permutation representation of M4(2)⋊6D4
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)
(2 6)(4 8)(9 13)(11 15)(18 22)(20 24)(26 30)(28 32)
(1 20 25 13)(2 19 26 12)(3 18 27 11)(4 17 28 10)(5 24 29 9)(6 23 30 16)(7 22 31 15)(8 21 32 14)
(1 22)(2 17)(3 24)(4 19)(5 18)(6 21)(7 20)(8 23)(9 27)(10 26)(11 29)(12 28)(13 31)(14 30)(15 25)(16 32)

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

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

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

Matrix representation of M4(2)⋊6D4 in GL6(𝔽17)

090000
200000
000010
000001
0001600
001000
,
1600000
0160000
001000
000100
0000160
0000016
,
0150000
900000
000001
000010
000100
001000
,
0150000
800000
000010
000001
001000
000100

G:=sub<GL(6,GF(17))| [0,2,0,0,0,0,9,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,1,0,0,0,0,0,0,1,0,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[0,9,0,0,0,0,15,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0],[0,8,0,0,0,0,15,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0] >;

M4(2)⋊6D4 in GAP, Magma, Sage, TeX

M_4(2)\rtimes_6D_4
% in TeX

G:=Group("M4(2):6D4");
// GroupNames label

G:=SmallGroup(128,769);
// by ID

G=gap.SmallGroup(128,769);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,141,422,387,58,2804,1411,718,172,4037]);
// 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^-1,d*a*d=a^3*b,c*b*c^-1=d*b*d=a^4*b,d*c*d=c^-1>;
// generators/relations

Export

Character table of M4(2)⋊6D4 in TeX

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