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G = D4.C8order 64 = 26

The non-split extension by D4 of C8 acting via C8/C4=C2

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

Aliases: D4.C8, Q8.C8, C8.25D4, M5(2)⋊4C2, M4(2).3C4, C22.1M4(2), (C2×C16)⋊2C2, C4.3(C2×C8), C8○D4.2C2, C4○D4.1C4, C2.8(C22⋊C8), (C2×C8).95C22, C4.30(C22⋊C4), (C2×C4).40(C2×C4), SmallGroup(64,31)

Series: Derived Chief Lower central Upper central Jennings

C1C4 — D4.C8
C1C2C4C8C2×C8C8○D4 — D4.C8
C1C2C4 — D4.C8
C1C8C2×C8 — D4.C8
C1C2C2C2C2C4C4C2×C8 — D4.C8

Generators and relations for D4.C8
 G = < a,b,c | a4=b2=1, c8=a2, bab=a-1, ac=ca, cbc-1=ab >

2C2
4C2
2C22
2C4
2D4
2C2×C4
2C8
2C16
2C16
2C2×C8
2M4(2)

Character table of D4.C8

 class 12A2B2C4A4B4C4D8A8B8C8D8E8F8G8H16A16B16C16D16E16F16G16H16I16J16K16L
 size 1124112411112244222222224444
ρ11111111111111111111111111111    trivial
ρ2111-1111-1111111-1-1-1-1-1-1-1-1-1-11111    linear of order 2
ρ31111111111111111-1-1-1-1-1-1-1-1-1-1-1-1    linear of order 2
ρ4111-1111-1111111-1-111111111-1-1-1-1    linear of order 2
ρ511111111-1-1-1-1-1-1-1-1ii-i-ii-i-iiii-i-i    linear of order 4
ρ611111111-1-1-1-1-1-1-1-1-i-iii-iii-i-i-iii    linear of order 4
ρ7111-1111-1-1-1-1-1-1-111ii-i-ii-i-ii-i-iii    linear of order 4
ρ8111-1111-1-1-1-1-1-1-111-i-iii-iii-iii-i-i    linear of order 4
ρ911-1-1-1-111i-ii-i-ii-iiζ87ζ87ζ8ζ8ζ83ζ85ζ85ζ83ζ87ζ83ζ8ζ85    linear of order 8
ρ1011-11-1-11-1-ii-iii-i-iiζ85ζ85ζ83ζ83ζ8ζ87ζ87ζ8ζ8ζ85ζ87ζ83    linear of order 8
ρ1111-11-1-11-1-ii-iii-i-iiζ8ζ8ζ87ζ87ζ85ζ83ζ83ζ85ζ85ζ8ζ83ζ87    linear of order 8
ρ1211-1-1-1-111i-ii-i-ii-iiζ83ζ83ζ85ζ85ζ87ζ8ζ8ζ87ζ83ζ87ζ85ζ8    linear of order 8
ρ1311-11-1-11-1i-ii-i-iii-iζ83ζ83ζ85ζ85ζ87ζ8ζ8ζ87ζ87ζ83ζ8ζ85    linear of order 8
ρ1411-1-1-1-111-ii-iii-ii-iζ85ζ85ζ83ζ83ζ8ζ87ζ87ζ8ζ85ζ8ζ83ζ87    linear of order 8
ρ1511-11-1-11-1i-ii-i-iii-iζ87ζ87ζ8ζ8ζ83ζ85ζ85ζ83ζ83ζ87ζ85ζ8    linear of order 8
ρ1611-1-1-1-111-ii-iii-ii-iζ8ζ8ζ87ζ87ζ85ζ83ζ83ζ85ζ8ζ85ζ87ζ83    linear of order 8
ρ1722-2022-202222-2-200000000000000    orthogonal lifted from D4
ρ1822-2022-20-2-2-2-22200000000000000    orthogonal lifted from D4
ρ192220-2-2-20-2i2i-2i2i-2i2i00000000000000    complex lifted from M4(2)
ρ202220-2-2-202i-2i2i-2i2i-2i00000000000000    complex lifted from M4(2)
ρ212-200-2i2i00161416216616100000ζ16151611ζ167163ζ1613169ζ16516ζ1611167ζ169165ζ161316ζ16151630000    complex faithful
ρ222-2002i-2i00162161416101660000ζ1613169ζ16516ζ16151611ζ167163ζ161316ζ1615163ζ1611167ζ1691650000    complex faithful
ρ232-200-2i2i00161416216616100000ζ167163ζ16151611ζ16516ζ1613169ζ1615163ζ161316ζ169165ζ16111670000    complex faithful
ρ242-2002i-2i00161016616216140000ζ169165ζ161316ζ1615163ζ1611167ζ1613169ζ167163ζ16151611ζ165160000    complex faithful
ρ252-200-2i2i00166161016141620000ζ1611167ζ1615163ζ161316ζ169165ζ167163ζ1613169ζ16516ζ161516110000    complex faithful
ρ262-2002i-2i00161016616216140000ζ161316ζ169165ζ1611167ζ1615163ζ16516ζ16151611ζ167163ζ16131690000    complex faithful
ρ272-2002i-2i00162161416101660000ζ16516ζ1613169ζ167163ζ16151611ζ169165ζ1611167ζ1615163ζ1613160000    complex faithful
ρ282-200-2i2i00166161016141620000ζ1615163ζ1611167ζ169165ζ161316ζ16151611ζ16516ζ1613169ζ1671630000    complex faithful

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

D4.C8 is a maximal subgroup of
Q16.10D4  Q16.D4  D8.3D4  D8.12D4  C8.7S4
 D4p.C8: C16○D8  D8.C8  D12.C8  Dic6.C8  D20.3C8  D20.4C8  D20.C8  D28.C8 ...
 (Cp×D4).C8: M5(2)⋊12C22  C24.99D4  C40.92D4  D4.(C5⋊C8)  C56.92D4 ...
D4.C8 is a maximal quotient of
C23.7M4(2)  Q8⋊C16  C8.17Q16  M5(2)⋊7C4  D20.C8
 C8.D4p: D4⋊C16  C8.31D8  D12.C8  Dic6.C8  D20.3C8  D20.4C8  D28.C8  Dic14.C8 ...
 (Cp×D4).C8: C23.M4(2)  C24.99D4  C40.92D4  D4.(C5⋊C8)  C56.92D4 ...

Matrix representation of D4.C8 in GL2(𝔽17) generated by

09
150
,
09
20
,
168
216
G:=sub<GL(2,GF(17))| [0,15,9,0],[0,2,9,0],[16,2,8,16] >;

D4.C8 in GAP, Magma, Sage, TeX

D_4.C_8
% in TeX

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

G:=SmallGroup(64,31);
// by ID

G=gap.SmallGroup(64,31);
# by ID

G:=PCGroup([6,-2,2,-2,2,-2,-2,48,73,332,158,489,69,88]);
// Polycyclic

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

Export

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

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