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G = C8.D4order 64 = 26

1st non-split extension by C8 of D4 acting via D4/C2=C22

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

Aliases: C8.1D4, C23.18D4, C4.Q85C2, (C2×Q16)⋊7C2, (C2×C4).32D4, C4.58(C2×D4), C22⋊Q8.4C2, Q8⋊C418C2, C4.12(C4○D4), C4⋊C4.10C22, (C2×C4).98C23, (C2×C8).53C22, C22.94(C2×D4), C2.22(C4⋊D4), (C2×M4(2)).3C2, (C2×Q8).14C22, C2.13(C8.C22), (C22×C4).50C22, SmallGroup(64,151)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C8.D4
C1C2C22C2×C4C22×C4C2×M4(2) — C8.D4
C1C2C2×C4 — C8.D4
C1C22C22×C4 — C8.D4
C1C2C2C2×C4 — C8.D4

Generators and relations for C8.D4
 G = < a,b,c | a8=b4=1, c2=a4, bab-1=a3, cac-1=a-1, cbc-1=a4b-1 >

4C2
2C22
2C4
2C22
2C22
4C4
4C4
4C4
4C4
2C8
2Q8
2C2×C4
2Q8
2C2×C4
2C2×C4
2Q8
2C2×C4
2C2×C4
2C2×C4
2Q8
2C4⋊C4
2C22⋊C4
2C22⋊C4
2Q16
2Q16
2M4(2)
2C4⋊C4
2M4(2)

Character table of C8.D4

 class 12A2B2C2D4A4B4C4D4E4F4G8A8B8C8D
 size 1111422488884444
ρ11111111111111111    trivial
ρ211111111-111-1-1-1-1-1    linear of order 2
ρ31111-111-111-1-11-11-1    linear of order 2
ρ41111-111-1-11-11-11-11    linear of order 2
ρ51111-111-11-11-1-11-11    linear of order 2
ρ61111-111-1-1-1111-11-1    linear of order 2
ρ7111111111-1-11-1-1-1-1    linear of order 2
ρ811111111-1-1-1-11111    linear of order 2
ρ92-22-202-200000-2020    orthogonal lifted from D4
ρ1022222-2-2-200000000    orthogonal lifted from D4
ρ112222-2-2-2200000000    orthogonal lifted from D4
ρ122-22-202-20000020-20    orthogonal lifted from D4
ρ132-22-20-220000002i0-2i    complex lifted from C4○D4
ρ142-22-20-22000000-2i02i    complex lifted from C4○D4
ρ154-4-44000000000000    symplectic lifted from C8.C22, Schur index 2
ρ1644-4-4000000000000    symplectic lifted from C8.C22, Schur index 2

Smallest permutation representation of C8.D4
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 14 19 32)(2 9 20 27)(3 12 21 30)(4 15 22 25)(5 10 23 28)(6 13 24 31)(7 16 17 26)(8 11 18 29)
(1 28 5 32)(2 27 6 31)(3 26 7 30)(4 25 8 29)(9 24 13 20)(10 23 14 19)(11 22 15 18)(12 21 16 17)

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

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

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

C8.D4 is a maximal subgroup of
C23.2SD16  C23.10SD16  C24.110D4  M4(2)⋊15D4  C8.D4⋊C2  (C2×C8)⋊13D4  M4(2)⋊17D4  C42.260D4  C42.262D4  C24.128D4  C24.129D4  C24.130D4  C4.162+ 1+4  C4.172+ 1+4  C4.192+ 1+4  C42.300D4  C42.303D4  C42.304D4
 C23.D4p: C23.2D8  D8.D4  C24.4D4  C40.4D4  C56.4D4 ...
 C4⋊C4.D2p: C42.385C23  C42.389C23  C42.390C23  C42.25C23  C42.28C23  C42.30C23  D810D4  Q169D4 ...
 (C2p×Q16)⋊C2: C42.256D4  C24.36D4  C40.36D4  C56.36D4 ...
C8.D4 is a maximal quotient of
 C8.D4p: C8.D8  C8.2D12  C8.2D20  C8.2D28 ...
 C23.D4p: C23.12D8  C24.4D4  C40.4D4  C56.4D4 ...
 C4⋊C4.D2p: C8.SD16  C8.8SD16  C8.3Q16  C42.254C23  C42.255C23  C232Q16  C24.85D4  (C2×Q8).8Q8 ...
 (C2×C8).D2p: C24.67D4  C24.75D4  C2.(C8⋊D4)  C8⋊(C4⋊C4)  (C2×Q16)⋊10C4  (C2×C4)⋊3Q16  (C2×C4).26D8  C24.36D4 ...

Matrix representation of C8.D4 in GL6(𝔽17)

1600000
0160000
00711150
00117015
0001106
0000610
,
16150000
110000
00143710
0012121010
007635
0065145
,
16150000
010000
00143710
0012121010
0021135
001110145

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,7,11,0,0,0,0,11,7,1,0,0,0,15,0,10,6,0,0,0,15,6,10],[16,1,0,0,0,0,15,1,0,0,0,0,0,0,14,12,7,6,0,0,3,12,6,5,0,0,7,10,3,14,0,0,10,10,5,5],[16,0,0,0,0,0,15,1,0,0,0,0,0,0,14,12,2,11,0,0,3,12,11,10,0,0,7,10,3,14,0,0,10,10,5,5] >;

C8.D4 in GAP, Magma, Sage, TeX

C_8.D_4
% in TeX

G:=Group("C8.D4");
// GroupNames label

G:=SmallGroup(64,151);
// by ID

G=gap.SmallGroup(64,151);
# by ID

G:=PCGroup([6,-2,2,2,-2,2,-2,192,121,247,362,332,963,117]);
// Polycyclic

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

Export

Subgroup lattice of C8.D4 in TeX
Character table of C8.D4 in TeX

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