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G = M4(2).12D4order 128 = 27

12nd non-split extension by M4(2) of D4 acting via D4/C2=C22

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

Aliases: M4(2).12D4, C4⋊C4.104D4, (C2×C8).160D4, (C2×D4).11Q8, C4.151(C4⋊D4), M4(2)⋊C47C2, C4.100(C22⋊Q8), C22.C4223C2, C22.5(C22⋊Q8), C2.28(D4.3D4), C2.21(D4.4D4), C23.280(C4○D4), (C22×C8).170C22, (C22×C4).735C23, C42.6C222C2, C23.37D4.9C2, (C22×D4).88C22, C2.4(C23.Q8), C22.241(C4⋊D4), C42⋊C2.66C22, C22.8(C422C2), (C2×M4(2)).27C22, (C2×C4).20(C2×Q8), (C2×C8.C4)⋊16C2, (C2×C4).1053(C2×D4), (C2×C4.D4).3C2, (C2×D4⋊C4).27C2, (C2×C4).350(C4○D4), (C2×C4⋊C4).139C22, SmallGroup(128,795)

Series: Derived Chief Lower central Upper central Jennings

C1C22×C4 — M4(2).12D4
C1C2C4C2×C4C22×C4C22×D4C2×D4⋊C4 — M4(2).12D4
C1C2C22×C4 — M4(2).12D4
C1C22C22×C4 — M4(2).12D4
C1C2C2C22×C4 — M4(2).12D4

Generators and relations for M4(2).12D4
 G = < a,b,c,d | a8=b2=c4=1, d2=a6b, bab=a5, cac-1=a-1, dad-1=a-1b, bc=cb, bd=db, dcd-1=a6bc-1 >

Subgroups: 280 in 114 conjugacy classes, 42 normal (38 characteristic)
C1, C2 [×3], C2 [×4], C4 [×4], C4 [×3], C22 [×3], C22 [×10], C8 [×6], C2×C4 [×6], C2×C4 [×4], D4 [×6], C23, C23 [×6], C42, C22⋊C4, C4⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×2], C2×C8 [×5], M4(2) [×2], M4(2) [×5], C22×C4, C22×C4, C2×D4 [×2], C2×D4 [×5], C24, C4.D4 [×2], D4⋊C4 [×4], C4⋊C8 [×2], C4.Q8, C2.D8, C8.C4 [×2], C2×C4⋊C4, C42⋊C2, C22×C8, C2×M4(2) [×3], C22×D4, C22.C42, C2×C4.D4, C2×D4⋊C4, C23.37D4, C42.6C22, M4(2)⋊C4, C2×C8.C4, M4(2).12D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], Q8 [×2], C23, C2×D4 [×3], C2×Q8, C4○D4 [×3], C4⋊D4 [×3], C22⋊Q8 [×3], C422C2, C23.Q8, D4.3D4, D4.4D4, M4(2).12D4

Character table of M4(2).12D4

 class 12A2B2C2D2E2F2G4A4B4C4D4E4F4G4H8A8B8C8D8E8F8G8H8I8J
 size 11112288222288884444888888
ρ111111111111111111111111111    trivial
ρ2111111-1-111111111-1-1-1-1-111-1-1-1    linear of order 2
ρ3111111-1-11111-1-111-1-1-1-11-1-1111    linear of order 2
ρ4111111111111-1-1111111-1-1-1-1-1-1    linear of order 2
ρ5111111111111-1-1-1-1-1-1-1-1111-11-1    linear of order 2
ρ6111111-1-11111-1-1-1-11111-1111-11    linear of order 2
ρ7111111-1-1111111-1-111111-1-1-11-1    linear of order 2
ρ811111111111111-1-1-1-1-1-1-1-1-11-11    linear of order 2
ρ92-2-222-2002-2-22000000000-22000    orthogonal lifted from D4
ρ102-2-222-2002-2-220000000002-2000    orthogonal lifted from D4
ρ112222-2-2002-22-22-2000000000000    orthogonal lifted from D4
ρ122222-2-200-22-2200002-22-2000000    orthogonal lifted from D4
ρ132222-2-200-22-220000-22-22000000    orthogonal lifted from D4
ρ142222-2-2002-22-2-22000000000000    orthogonal lifted from D4
ρ152-2-222-2-22-222-200000000000000    symplectic lifted from Q8, Schur index 2
ρ162-2-222-22-2-222-200000000000000    symplectic lifted from Q8, Schur index 2
ρ172-2-22-2200-2-22200000000000-2i02i    complex lifted from C4○D4
ρ182-2-22-2200-2-222000000000002i0-2i    complex lifted from C4○D4
ρ1922222200-2-2-2-2000000002i000-2i0    complex lifted from C4○D4
ρ202-2-22-220022-2-200-2i2i0000000000    complex lifted from C4○D4
ρ212-2-22-220022-2-2002i-2i0000000000    complex lifted from C4○D4
ρ2222222200-2-2-2-200000000-2i0002i0    complex lifted from C4○D4
ρ2344-4-4000000000000-220220000000    orthogonal lifted from D4.4D4
ρ2444-4-4000000000000220-220000000    orthogonal lifted from D4.4D4
ρ254-44-40000000000000-2-202-2000000    complex lifted from D4.3D4
ρ264-44-400000000000002-20-2-2000000    complex lifted from D4.3D4

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

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

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

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

Matrix representation of M4(2).12D4 in GL6(𝔽17)

100000
010000
0000160
0077162
000100
001010710
,
100000
010000
0016000
0001600
000010
007701
,
010000
1600000
00121200
0012500
0011010
001050
,
010000
100000
00121200
0051200
00161607
0016057

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,7,0,10,0,0,0,7,1,10,0,0,16,16,0,7,0,0,0,2,0,10],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,7,0,0,0,16,0,7,0,0,0,0,1,0,0,0,0,0,0,1],[0,16,0,0,0,0,1,0,0,0,0,0,0,0,12,12,1,1,0,0,12,5,1,0,0,0,0,0,0,5,0,0,0,0,10,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,12,5,16,16,0,0,12,12,16,0,0,0,0,0,0,5,0,0,0,0,7,7] >;

M4(2).12D4 in GAP, Magma, Sage, TeX

M_4(2)._{12}D_4
% in TeX

G:=Group("M4(2).12D4");
// GroupNames label

G:=SmallGroup(128,795);
// by ID

G=gap.SmallGroup(128,795);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,168,141,64,422,387,2019,521,248,2804,4037,1027,124]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=c^4=1,d^2=a^6*b,b*a*b=a^5,c*a*c^-1=a^-1,d*a*d^-1=a^-1*b,b*c=c*b,b*d=d*b,d*c*d^-1=a^6*b*c^-1>;
// generators/relations

Export

Character table of M4(2).12D4 in TeX

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