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G = C8.17D4order 64 = 26

4th non-split extension by C8 of D4 acting via D4/C22=C2

p-group, metabelian, nilpotent (class 4), monomial

Aliases: C8.17D4, C4.12D8, Q16.2C4, M5(2).3C2, C22.4SD16, C8.3(C2×C4), (C2×C4).12D4, (C2×Q16).5C2, C8.C4.1C2, C4.6(C22⋊C4), (C2×C8).11C22, C2.11(D4⋊C4), SmallGroup(64,43)

Series: Derived Chief Lower central Upper central Jennings

C1C8 — C8.17D4
C1C2C4C2×C4C2×C8C2×Q16 — C8.17D4
C1C2C4C8 — C8.17D4
C1C2C2×C4C2×C8 — C8.17D4
C1C2C2C2C2C4C4C2×C8 — C8.17D4

Generators and relations for C8.17D4
 G = < a,b,c | a8=1, b4=a4, c2=bab-1=a-1, ac=ca, cbc-1=a-1b3 >

2C2
4C4
4C4
2Q8
2Q8
4Q8
4C2×C4
4C8
2C16
2M4(2)
2Q16
2C2×Q8

Character table of C8.17D4

 class 12A2B4A4B4C4D8A8B8C8D8E16A16B16C16D
 size 1122288224884444
ρ11111111111111111    trivial
ρ211111-1-111111-1-1-1-1    linear of order 2
ρ31111111111-1-1-1-1-1-1    linear of order 2
ρ411111-1-1111-1-11111    linear of order 2
ρ511-11-11-1-1-11i-ii-ii-i    linear of order 4
ρ611-11-1-11-1-11i-i-ii-ii    linear of order 4
ρ711-11-11-1-1-11-ii-ii-ii    linear of order 4
ρ811-11-1-11-1-11-iii-ii-i    linear of order 4
ρ922-22-20022-2000000    orthogonal lifted from D4
ρ102222200-2-2-2000000    orthogonal lifted from D4
ρ1122-2-220000000-222-2    orthogonal lifted from D8
ρ1222-2-2200000002-2-22    orthogonal lifted from D8
ρ13222-2-20000000--2--2-2-2    complex lifted from SD16
ρ14222-2-20000000-2-2--2--2    complex lifted from SD16
ρ154-400000-22220000000    symplectic faithful, Schur index 2
ρ164-40000022-220000000    symplectic faithful, Schur index 2

Smallest permutation representation of C8.17D4
On 32 points
Generators in S32
(1 15 13 11 9 7 5 3)(2 16 14 12 10 8 6 4)(17 31 29 27 25 23 21 19)(18 32 30 28 26 24 22 20)
(1 22 13 26 9 30 5 18)(2 25 6 21 10 17 14 29)(3 20 15 24 11 28 7 32)(4 23 8 19 12 31 16 27)
(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)

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

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

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

C8.17D4 is a maximal subgroup of
C23.21SD16  Q32⋊C4  Q16.D4  D8.12D4  D4.4D8  D4.5D8  C23.10SD16  Dic20.C4
 C4p.D8: C8.3D8  C8.5D8  C24.8D4  C12.4D8  Q16.Dic3  C40.8D4  C20.4D8  Q16.Dic5 ...
C8.17D4 is a maximal quotient of
Q16⋊C8  C23.13SD16  C4.10D16  C8.2C42  Dic20.C4
 C4p.D8: C8.27D8  C24.8D4  C12.4D8  Q16.Dic3  C40.8D4  C20.4D8  Q16.Dic5  C56.8D4 ...

Matrix representation of C8.17D4 in GL4(𝔽7) generated by

2051
1221
1665
5515
,
4055
6225
1105
0341
,
0255
2201
6144
2121
G:=sub<GL(4,GF(7))| [2,1,1,5,0,2,6,5,5,2,6,1,1,1,5,5],[4,6,1,0,0,2,1,3,5,2,0,4,5,5,5,1],[0,2,6,2,2,2,1,1,5,0,4,2,5,1,4,1] >;

C8.17D4 in GAP, Magma, Sage, TeX

C_8._{17}D_4
% in TeX

G:=Group("C8.17D4");
// GroupNames label

G:=SmallGroup(64,43);
// by ID

G=gap.SmallGroup(64,43);
# by ID

G:=PCGroup([6,-2,2,-2,2,-2,-2,48,73,199,362,476,86,489,117,1444,730,88]);
// Polycyclic

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

Export

Subgroup lattice of C8.17D4 in TeX
Character table of C8.17D4 in TeX

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