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G = D4.3D4order 64 = 26

3rd non-split extension by D4 of D4 acting via D4/C4=C2

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

Aliases: D4.3D4, C8.23D4, Q8.3D4, M4(2).3C22, C8○D41C2, C8⋊C22.C2, C4.35(C2×D4), C8.C45C2, (C2×SD16)⋊2C2, C4.D43C2, C8.C223C2, (C2×C4).8C23, C4.10D43C2, (C2×C8).17C22, C4○D4.9C22, C2.23(C4⋊D4), (C2×D4).19C22, C22.6(C4○D4), (C2×Q8).15C22, SmallGroup(64,152)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — D4.3D4
C1C2C4C2×C4C4○D4C8○D4 — D4.3D4
C1C2C2×C4 — D4.3D4
C1C2C2×C4 — D4.3D4
C1C2C2C2×C4 — D4.3D4

Generators and relations for D4.3D4
 G = < a,b,c,d | a4=b2=1, c4=d2=a2, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=c3 >

2C2
4C2
8C2
2C22
2C4
4C4
4C22
8C22
2C23
2D4
2C8
2C2×C4
2Q8
2C8
2Q8
2C8
2D4
2D4
2C2×C4
2SD16
2D8
2SD16
2SD16
2M4(2)
2Q16
2SD16
2C2×C8

Character table of D4.3D4

 class 12A2B2C2D4A4B4C4D8A8B8C8D8E8F8G
 size 1124822482244488
ρ11111111111111111    trivial
ρ2111-1-111-1-111-11-111    linear of order 2
ρ311111111-1-1-1-1-1-11-1    linear of order 2
ρ4111-1-111-11-1-11-111-1    linear of order 2
ρ5111-1111-1111-11-1-1-1    linear of order 2
ρ61111-1111-111111-1-1    linear of order 2
ρ7111-1111-1-1-1-11-11-11    linear of order 2
ρ81111-11111-1-1-1-1-1-11    linear of order 2
ρ922-200-2200-2-202000    orthogonal lifted from D4
ρ1022-2202-2-200000000    orthogonal lifted from D4
ρ1122-200-2200220-2000    orthogonal lifted from D4
ρ1222-2-202-2200000000    orthogonal lifted from D4
ρ1322200-2-200002i0-2i00    complex lifted from C4○D4
ρ1422200-2-20000-2i02i00    complex lifted from C4○D4
ρ154-40000000-2-22-200000    complex faithful
ρ164-400000002-2-2-200000    complex faithful

Permutation representations of D4.3D4
On 16 points - transitive group 16T132
Generators in S16
(1 7 5 3)(2 8 6 4)(9 11 13 15)(10 12 14 16)
(1 12)(2 13)(3 14)(4 15)(5 16)(6 9)(7 10)(8 11)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(1 4 5 8)(2 7 6 3)(9 12 13 16)(10 15 14 11)

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

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

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

G:=TransitiveGroup(16,132);

D4.3D4 is a maximal subgroup of
M4(2).10C23  D811D4  D86D4  C8.4S4
 D4p.D4: D8.13D4  D8○SD16  D12.2D4  D12.6D4  C24.42D4  C24.44D4  D20.2D4  D20.6D4 ...
 M4(2).D2p: M4(2).37D4  M4(2).38D4  Q8.8D12  M4(2).13D6  M4(2).15D6  D4.3D20  M4(2).13D10  M4(2).15D10 ...
D4.3D4 is a maximal quotient of
 M4(2).D2p: M4(2).48D4  M4(2).49D4  M4(2).31D4  M4(2).5D4  M4(2).10D4  M4(2).11D4  M4(2).12D4  M4(2).13D4 ...
 (C2×C8).D2p: C88D8  C814SD16  C811SD16  C88Q16  D4.2D8  Q8.2D8  D4.Q16  Q8.2Q16 ...

Matrix representation of D4.3D4 in GL4(𝔽3) generated by

1020
0101
2020
0102
,
1012
0112
1220
1202
,
0202
1220
0102
1012
,
1202
2110
2022
0122
G:=sub<GL(4,GF(3))| [1,0,2,0,0,1,0,1,2,0,2,0,0,1,0,2],[1,0,1,1,0,1,2,2,1,1,2,0,2,2,0,2],[0,1,0,1,2,2,1,0,0,2,0,1,2,0,2,2],[1,2,2,0,2,1,0,1,0,1,2,2,2,0,2,2] >;

D4.3D4 in GAP, Magma, Sage, TeX

D_4._3D_4
% in TeX

G:=Group("D4.3D4");
// GroupNames label

G:=SmallGroup(64,152);
// by ID

G=gap.SmallGroup(64,152);
# by ID

G:=PCGroup([6,-2,2,2,-2,2,-2,121,55,362,963,117,1444,376,88]);
// Polycyclic

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

Export

Subgroup lattice of D4.3D4 in TeX
Character table of D4.3D4 in TeX

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