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G = M4(2).8D4order 128 = 27

8th non-split extension by M4(2) of D4 acting via D4/C2=C22

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

Aliases: M4(2).8D4, C4.70C22≀C2, (C2×D4).109D4, (C2×Q8).100D4, C4.28(C4⋊D4), M4(2).C44C2, C42⋊C223C2, (C22×C4).41C23, M4(2)⋊4C410C2, C23.131(C4○D4), C23.37D430C2, C22.69(C4⋊D4), (C22×D4).84C22, C42⋊C2.61C22, C22.10(C4.4D4), C4.23(C22.D4), C2.21(C23.10D4), (C2×M4(2)).230C22, M4(2).8C2212C2, C22.57(C22.D4), (C2×C4.D4)⋊6C2, (C2×C4).261(C2×D4), (C2×C8⋊C22).7C2, (C2×C4).344(C4○D4), (C2×C4○D4).61C22, SmallGroup(128,780)

Series: Derived Chief Lower central Upper central Jennings

C1C22×C4 — M4(2).8D4
C1C2C22C2×C4C22×C4C2×M4(2)M4(2).8C22 — M4(2).8D4
C1C2C22×C4 — M4(2).8D4
C1C2C22×C4 — M4(2).8D4
C1C2C2C22×C4 — M4(2).8D4

Generators and relations for M4(2).8D4
 G = < a,b,c,d | a8=b2=d2=1, c4=a4, bab=a5, cac-1=a3, dad=ab, cbc-1=a4b, bd=db, dcd=a4c3 >

Subgroups: 344 in 137 conjugacy classes, 40 normal (34 characteristic)
C1, C2, C2 [×6], C4 [×4], C4 [×2], C22 [×3], C22 [×11], C8 [×6], C2×C4 [×6], C2×C4 [×4], D4 [×10], Q8 [×2], C23, C23 [×7], C42, C22⋊C4, C4⋊C4, C2×C8 [×4], M4(2) [×4], M4(2) [×6], D8 [×4], SD16 [×4], C22×C4, C22×C4, C2×D4, C2×D4 [×2], C2×D4 [×6], C2×Q8, C4○D4 [×4], C24, C4.D4 [×3], C4.10D4, D4⋊C4 [×2], C4≀C2 [×2], C8.C4 [×2], C42⋊C2, C2×M4(2) [×4], C2×D8, C2×SD16, C8⋊C22 [×4], C22×D4, C2×C4○D4, M4(2)⋊4C4, C2×C4.D4, M4(2).8C22, C23.37D4, C42⋊C22, M4(2).C4, C2×C8⋊C22, M4(2).8D4
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, M4(2).8D4

Character table of M4(2).8D4

 class 12A2B2C2D2E2F2G4A4B4C4D4E4F4G8A8B8C8D8E8F8G8H
 size 11222888222288888888888
ρ111111111111111111111111    trivial
ρ211111-1-1-11111-1-1-1-11-111111    linear of order 2
ρ311111-111111111-1-1-1-1-11-1-11    linear of order 2
ρ4111111-1-11111-1-111-11-11-1-11    linear of order 2
ρ511111-1111111-1-1-11-111-1-11-1    linear of order 2
ρ6111111-1-11111111-1-1-11-1-11-1    linear of order 2
ρ7111111111111-1-11-11-1-1-11-1-1    linear of order 2
ρ811111-1-1-1111111-1111-1-11-1-1    linear of order 2
ρ922-22-22002-22-200-200000000    orthogonal lifted from D4
ρ10222-2-2000-222-20000-2000200    orthogonal lifted from D4
ρ1122-22-2-2002-22-200200000000    orthogonal lifted from D4
ρ1222-2-2200022-2-2000000200-20    orthogonal lifted from D4
ρ13222-2-2000-222-200002000-200    orthogonal lifted from D4
ρ1422-2-220-22-2-22200000000000    orthogonal lifted from D4
ρ1522-2-2200022-2-2000000-20020    orthogonal lifted from D4
ρ1622-2-2202-2-2-22200000000000    orthogonal lifted from D4
ρ1722222000-2-2-2-2000-2i02i00000    complex lifted from C4○D4
ρ1822-22-2000-22-2200000002i00-2i    complex lifted from C4○D4
ρ1922222000-2-2-2-20002i0-2i00000    complex lifted from C4○D4
ρ20222-2-20002-2-22-2i2i000000000    complex lifted from C4○D4
ρ2122-22-2000-22-220000000-2i002i    complex lifted from C4○D4
ρ22222-2-20002-2-222i-2i000000000    complex lifted from C4○D4
ρ238-8000000000000000000000    orthogonal faithful

Permutation representations of M4(2).8D4
On 16 points - transitive group 16T411
Generators in S16
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(2 6)(4 8)(10 14)(12 16)
(1 12 3 10 5 16 7 14)(2 15 4 13 6 11 8 9)
(2 6)(3 7)(9 11)(10 16)(12 14)(13 15)

G:=sub<Sym(16)| (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (2,6)(4,8)(10,14)(12,16), (1,12,3,10,5,16,7,14)(2,15,4,13,6,11,8,9), (2,6)(3,7)(9,11)(10,16)(12,14)(13,15)>;

G:=Group( (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (2,6)(4,8)(10,14)(12,16), (1,12,3,10,5,16,7,14)(2,15,4,13,6,11,8,9), (2,6)(3,7)(9,11)(10,16)(12,14)(13,15) );

G=PermutationGroup([(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(2,6),(4,8),(10,14),(12,16)], [(1,12,3,10,5,16,7,14),(2,15,4,13,6,11,8,9)], [(2,6),(3,7),(9,11),(10,16),(12,14),(13,15)])

G:=TransitiveGroup(16,411);

Matrix representation of M4(2).8D4 in GL8(ℤ)

00000010
0000000-1
0000-1000
00000100
000-10000
00-100000
0-1000000
-10000000
,
10000000
01000000
00100000
00010000
0000-1000
00000-100
000000-10
0000000-1
,
00000-100
0000-1000
00000010
0000000-1
10000000
0-1000000
00010000
00100000
,
10000000
0-1000000
00010000
00100000
00000-100
0000-1000
00000010
0000000-1

G:=sub<GL(8,Integers())| [0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1],[0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0],[1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1] >;

M4(2).8D4 in GAP, Magma, Sage, TeX

M_4(2)._8D_4
% in TeX

G:=Group("M4(2).8D4");
// GroupNames label

G:=SmallGroup(128,780);
// by ID

G=gap.SmallGroup(128,780);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,141,422,387,58,2019,1018,248,2804,172,1027]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=d^2=1,c^4=a^4,b*a*b=a^5,c*a*c^-1=a^3,d*a*d=a*b,c*b*c^-1=a^4*b,b*d=d*b,d*c*d=a^4*c^3>;
// generators/relations

Export

Character table of M4(2).8D4 in TeX

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