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G = D8.C4order 64 = 26

1st non-split extension by D8 of C4 acting via C4/C2=C2

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

Aliases: D8.1C4, C8.21D4, C4.18D8, Q16.1C4, C22.1SD16, (C2×C16)⋊4C2, C8.9(C2×C4), C4○D8.1C2, (C2×C4).62D4, C8.C41C2, C4.3(C22⋊C4), (C2×C8).86C22, C2.8(D4⋊C4), SmallGroup(64,40)

Series: Derived Chief Lower central Upper central Jennings

C1C8 — D8.C4
C1C2C4C2×C4C2×C8C4○D8 — D8.C4
C1C2C4C8 — D8.C4
C1C4C2×C4C2×C8 — D8.C4
C1C2C2C2C2C4C4C2×C8 — D8.C4

Generators and relations for D8.C4
 G = < a,b,c | a8=b2=1, c4=a4, bab=cac-1=a-1, cbc-1=a5b >

2C2
8C2
4C22
4C4
2Q8
2D4
4C2×C4
4D4
4C8
2C16
2M4(2)
2SD16
2C4○D4

Character table of D8.C4

 class 12A2B2C4A4B4C4D8A8B8C8D8E8F16A16B16C16D16E16F16G16H
 size 1128112822228822222222
ρ11111111111111111111111    trivial
ρ2111-1111-11111-1-111111111    linear of order 2
ρ3111111111111-1-1-1-1-1-1-1-1-1-1    linear of order 2
ρ4111-1111-1111111-1-1-1-1-1-1-1-1    linear of order 2
ρ511-11-1-11-1-111-1-iii-i-iiii-i-i    linear of order 4
ρ611-1-1-1-111-111-1-ii-iii-i-i-iii    linear of order 4
ρ711-1-1-1-111-111-1i-ii-i-iiii-i-i    linear of order 4
ρ811-11-1-11-1-111-1i-i-iii-i-i-iii    linear of order 4
ρ922-20-2-2202-2-220000000000    orthogonal lifted from D4
ρ1022202220-2-2-2-20000000000    orthogonal lifted from D4
ρ1122-2022-200000002-22-22-22-2    orthogonal lifted from D8
ρ1222-2022-20000000-22-22-22-22    orthogonal lifted from D8
ρ132220-2-2-20000000--2--2-2-2--2-2-2--2    complex lifted from SD16
ρ142220-2-2-20000000-2-2--2--2-2--2--2-2    complex lifted from SD16
ρ152-2002i-2i00-22-2--200ζ1615165ζ16151613ζ161116ζ16316ζ1613167ζ1611169ζ169163ζ167165    complex faithful
ρ162-2002i-2i00--2-22-200ζ1611169ζ161116ζ167165ζ1615165ζ16316ζ1613167ζ16151613ζ169163    complex faithful
ρ172-2002i-2i00-22-2--200ζ1613167ζ167165ζ169163ζ1611169ζ1615165ζ16316ζ161116ζ16151613    complex faithful
ρ182-200-2i2i00--22-2-200ζ161116ζ16316ζ1615165ζ16151613ζ169163ζ167165ζ1613167ζ1611169    complex faithful
ρ192-200-2i2i00--22-2-200ζ169163ζ1611169ζ1613167ζ167165ζ161116ζ16151613ζ1615165ζ16316    complex faithful
ρ202-2002i-2i00--2-22-200ζ16316ζ169163ζ16151613ζ1613167ζ1611169ζ1615165ζ167165ζ161116    complex faithful
ρ212-200-2i2i00-2-22--200ζ167165ζ1615165ζ1611169ζ161116ζ16151613ζ169163ζ16316ζ1613167    complex faithful
ρ222-200-2i2i00-2-22--200ζ16151613ζ1613167ζ16316ζ169163ζ167165ζ161116ζ1611169ζ1615165    complex faithful

Smallest permutation representation of D8.C4
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 8)(2 7)(3 6)(4 5)(9 13)(10 12)(14 16)(17 24)(18 23)(19 22)(20 21)(25 29)(26 28)(30 32)
(1 12 21 32 5 16 17 28)(2 11 22 31 6 15 18 27)(3 10 23 30 7 14 19 26)(4 9 24 29 8 13 20 25)

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

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

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

D8.C4 is a maximal subgroup of
C23.20SD16  Q16.10D4  Q16.D4  D8.3D4  D8.12D4  D8.F5  Q16.F5
 C4p.D8: C8○D16  D165C4  Dic12.C4  D24.1C4  C24.41D4  D40.5C4  D40.3C4  C20.58D8 ...
D8.C4 is a maximal quotient of
C4.16D16  Q161C8  C23.12SD16  C23.13SD16  C8.16Q16  C8.9C42  D8.F5  Q16.F5
 C4p.D8: C8.30D8  Dic12.C4  D24.1C4  C24.41D4  D40.5C4  D40.3C4  C20.58D8  Dic28.C4 ...

Matrix representation of D8.C4 in GL2(𝔽17) generated by

143
1414
,
143
33
,
1514
142
G:=sub<GL(2,GF(17))| [14,14,3,14],[14,3,3,3],[15,14,14,2] >;

D8.C4 in GAP, Magma, Sage, TeX

D_8.C_4
% in TeX

G:=Group("D8.C4");
// GroupNames label

G:=SmallGroup(64,40);
// by ID

G=gap.SmallGroup(64,40);
# by ID

G:=PCGroup([6,-2,2,-2,2,-2,-2,48,73,362,188,230,117,1444,730,88]);
// Polycyclic

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

Export

Subgroup lattice of D8.C4 in TeX
Character table of D8.C4 in TeX

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