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G = D4.7D4order 64 = 26

2nd non-split extension by D4 of D4 acting via D4/C22=C2

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

Aliases: D4.7D4, Q8.7D4, C23.15D4, C22⋊C86C2, (C2×Q16)⋊2C2, C4.24(C2×D4), C22⋊Q82C2, D4⋊C45C2, C2.7(C4○D8), (C2×C4).105D4, C4⋊C4.4C22, Q8⋊C410C2, C2.13C22≀C2, (C2×SD16)⋊10C2, (C2×C8).28C22, (C2×C4).86C23, C22.82(C2×D4), (C2×Q8).4C22, (C2×D4).55C22, C2.8(C8.C22), (C22×C4).47C22, (C2×C4○D4).8C2, SmallGroup(64,133)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — D4.7D4
C1C2C22C2×C4C22×C4C2×C4○D4 — D4.7D4
C1C2C2×C4 — D4.7D4
C1C22C22×C4 — D4.7D4
C1C2C2C2×C4 — D4.7D4

Generators and relations for D4.7D4
 G = < a,b,c,d | a4=b2=c4=1, d2=a2, bab=cac-1=dad-1=a-1, cbc-1=dbd-1=ab, dcd-1=c-1 >

Subgroups: 137 in 76 conjugacy classes, 29 normal (25 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×5], C22, C22 [×7], C8 [×2], C2×C4 [×2], C2×C4 [×9], D4 [×2], D4 [×5], Q8 [×2], Q8 [×3], C23, C23, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8 [×2], SD16 [×2], Q16 [×2], C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8 [×2], C4○D4 [×4], C22⋊C8, D4⋊C4, Q8⋊C4, C22⋊Q8, C2×SD16, C2×Q16, C2×C4○D4, D4.7D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, C2×D4 [×3], C22≀C2, C4○D8, C8.C22, D4.7D4

Character table of D4.7D4

 class 12A2B2C2D2E2F4A4B4C4D4E4F4G4H8A8B8C8D
 size 1111444222244884444
ρ11111111111111111111    trivial
ρ211111-111-11-1-1-1-11-111-1    linear of order 2
ρ31111-1-1-11-11-1111-1-111-1    linear of order 2
ρ41111-11-11111-1-1-1-11111    linear of order 2
ρ511111-111-11-1-1-11-11-1-11    linear of order 2
ρ61111111111111-1-1-1-1-1-1    linear of order 2
ρ71111-11-11111-1-111-1-1-1-1    linear of order 2
ρ81111-1-1-11-11-111-111-1-11    linear of order 2
ρ92-22-200020-202-2000000    orthogonal lifted from D4
ρ102-22-220-2-202000000000    orthogonal lifted from D4
ρ1122220-20-22-2200000000    orthogonal lifted from D4
ρ122222020-2-2-2-200000000    orthogonal lifted from D4
ρ132-22-200020-20-22000000    orthogonal lifted from D4
ρ142-22-2-202-202000000000    orthogonal lifted from D4
ρ152-2-220000-2i02i0000--22-2-2    complex lifted from C4○D8
ρ162-2-2200002i0-2i0000-22-2--2    complex lifted from C4○D8
ρ172-2-220000-2i02i0000-2-22--2    complex lifted from C4○D8
ρ182-2-2200002i0-2i0000--2-22-2    complex lifted from C4○D8
ρ1944-4-4000000000000000    symplectic lifted from C8.C22, Schur index 2

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

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

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

D4.7D4 is a maximal subgroup of
C42.229D4  C42.234D4  C24.123D4  C24.129D4  C42.270D4  C42.274D4  SL2(𝔽3).D4
 (Cp×D4).D4: D4.(C2×D4)  Q8.(C2×D4)  (C2×Q8)⋊17D4  C4.162+ 1+4  C4.182+ 1+4  C4.192+ 1+4  SD166D4  D810D4 ...
 C8pD4⋊C2: C24.124D4  C24.130D4  C42.268D4  C42.277D4  C42.408C23  C42.410C23 ...
 C4⋊C4.D2p: C42.354C23  C42.358C23  C42.359C23  C42.409C23  C42.411C23  Q8.11D12  D12.37D4  Q8.D20 ...
 (C2p×Q16)⋊C2: C42.384D4  C42.451D4  D12.17D4  D20.17D4  D28.17D4 ...
 C2p.C22≀C2: C24.103D4  C24.104D4  C24.106D4  (C2×D4)⋊21D4  D12.32D4  D20.32D4  D28.32D4 ...
D4.7D4 is a maximal quotient of
D12.32D4  D12.17D4  D20.32D4  D20.17D4  D28.32D4  D28.17D4
 Q8.D4p: Q8.D8  Q8.11D12  Q8.D20  Q8.D28 ...
 D4.D4p: D4.7D8  D4.D12  D4.D20  D4.D28 ...
 (Cp×D4).D4: C4⋊C4.6D4  C24.12D4  C4⋊C4.18D4  C24.18D4  C42.191C23  Q82SD16  D4⋊Q16  C42.195C23 ...
 C4⋊C4.D2p: Q8.Q16  Q8.SD16  C24.71D4  Q8⋊C4⋊C4  C24.73D4  (C2×C4)⋊9Q16  C232Q16  C4⋊C4.85D4 ...

Matrix representation of D4.7D4 in GL4(𝔽17) generated by

1000
0100
0001
00160
,
1000
0100
0001
0010
,
12300
14500
00143
0033
,
14500
12300
00125
0055
G:=sub<GL(4,GF(17))| [1,0,0,0,0,1,0,0,0,0,0,16,0,0,1,0],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0],[12,14,0,0,3,5,0,0,0,0,14,3,0,0,3,3],[14,12,0,0,5,3,0,0,0,0,12,5,0,0,5,5] >;

D4.7D4 in GAP, Magma, Sage, TeX

D_4._7D_4
% in TeX

G:=Group("D4.7D4");
// GroupNames label

G:=SmallGroup(64,133);
// by ID

G=gap.SmallGroup(64,133);
# by ID

G:=PCGroup([6,-2,2,2,-2,2,-2,192,121,362,158,963,489,117]);
// Polycyclic

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

Export

Character table of D4.7D4 in TeX

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