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G = D8⋊2C4  order 64 = 26

2nd semidirect product of D8 and C4 acting via C4/C2=C2

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

Aliases: D8⋊2C4, Q16⋊2C4, C8.22D4, C4.8SD16, C22.3D8, M5(2)⋊5C2, C8.1(C2×C4), C4.Q8⋊1C2, C4○D8.2C2, (C2×C4).10D4, (C2×C8).9C22, C4.4(C22⋊C4), C2.9(D4⋊C4), SmallGroup(64,41)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C8 — D8⋊2C4
Chief series
C1 — C2 — C4 — C2×C4 — C2×C8 — C4○D8 — D8⋊2C4
Lower central
C1 — C2 — C4 — C8 — D8⋊2C4
Upper central
C1 — C2 — C2×C4 — C2×C8 — D8⋊2C4
Jennings
C1 — C2 — C2 — C2 — C2 — C4 — C4 — C2×C8 — D8⋊2C4

Generators and relations for D8⋊2C4
 G = < a,b,c | a8=b2=c4=1, bab=a-1, cac-1=a3, cbc-1=a5b >

C1
C2
2C2
8C2
C22
C4
C4
4C22
4C4
8C4
C8
C8
C2×C4
2D4
2Q8
4D4
4C2×C4
4C2×C4
C2×C8
Q16
D8
2C16
2SD16
2C4○D4
2C4⋊C4
C4.Q8
M5(2)
C4○D8
D8⋊2C4

Character table of D8⋊2C4

 class 12A2B2C4A4B4C4D4E8A8B8C16A16B16C16D
 size 1128228882244444
ρ11111111111111111    trivial
ρ2111-111-1-1-11111111    linear of order 2
ρ3111111-1-11111-1-1-1-1    linear of order 2
ρ4111-11111-1111-1-1-1-1    linear of order 2
ρ511-111-1-ii-1-1-11i-ii-i    linear of order 4
ρ611-111-1i-i-1-1-11-ii-ii    linear of order 4
ρ711-1-11-1i-i1-1-11i-ii-i    linear of order 4
ρ811-1-11-1-ii1-1-11-ii-ii    linear of order 4
ρ922-202-200022-20000    orthogonal lifted from D4
ρ10222022000-2-2-20000    orthogonal lifted from D4
ρ112220-2-2000000-√2√2√2-√2    orthogonal lifted from D8
ρ122220-2-2000000√2-√2-√2√2    orthogonal lifted from D8
ρ1322-20-22000000√-2√-2-√-2-√-2    complex lifted from SD16
ρ1422-20-22000000-√-2-√-2√-2√-2    complex lifted from SD16
ρ154-40000000-2√-22√-200000    complex faithful
ρ164-400000002√-2-2√-200000    complex faithful

Permutation representations of D8⋊2C4
►On 16 points - transitive group 16T156
Generators in S16
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)

(1 16)(2 15)(3 14)(4 13)(5 12)(6 11)(7 10)(8 9)

(2 4)(3 7)(6 8)(9 10 13 14)(11 16 15 12)
 
G:=sub<Sym(16)| (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,16)(2,15)(3,14)(4,13)(5,12)(6,11)(7,10)(8,9), (2,4)(3,7)(6,8)(9,10,13,14)(11,16,15,12)>;
 
G:=Group( (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,16)(2,15)(3,14)(4,13)(5,12)(6,11)(7,10)(8,9), (2,4)(3,7)(6,8)(9,10,13,14)(11,16,15,12) );
 
G=PermutationGroup([[(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(1,16),(2,15),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)], [(2,4),(3,7),(6,8),(9,10,13,14),(11,16,15,12)]])
 
G:=TransitiveGroup(16,156);
 

D8⋊2C4 is a maximal subgroup of
 C23.13D8  Q32⋊C4  D8⋊D4  D8.D4  D10.D8
 D8p⋊C4: D16⋊C4  D24⋊8C4  D24⋊2C4  D40⋊14C4  D40⋊8C4  D40⋊1C4  D56⋊8C4  D56⋊2C4 ...
 C4p.SD16: D8⋊3Q8  D8.2Q8  D8⋊2Dic3  D8⋊2Dic5  D8⋊2Dic7 ...
D8⋊2C4 is a maximal quotient of
 D8⋊C8  Q16⋊C8  C22.SD32  C23.32D8  C8.C42  D10.D8  D40⋊1C4
 C8.D4p: C8.30D8  D24⋊8C4  D40⋊14C4  D56⋊8C4 ...
 C4p.SD16: C8.16Q16  D24⋊2C4  D8⋊2Dic3  D40⋊8C4  D8⋊2Dic5  D56⋊2C4  D8⋊2Dic7 ...

Matrix representation of D8⋊2C4 ►in GL4(𝔽3) generated by

2002
0010
0110
2000
,
0020
1001
2000
0110
,
1002
0020
0100
0002
G:=sub<GL(4,GF(3))| [2,0,0,2,0,0,1,0,0,1,1,0,2,0,0,0],[0,1,2,0,0,0,0,1,2,0,0,1,0,1,0,0],[1,0,0,0,0,0,1,0,0,2,0,0,2,0,0,2] >;
 

D8⋊2C4 in GAP, Magma, Sage, TeX 

D_8\rtimes_2C_4
 
% in TeX
 
G:=Group("D8:2C4");
 
// GroupNames label
 
G:=SmallGroup(64,41);
 
// by ID
 
G=gap.SmallGroup(64,41);
 
# by ID
 
G:=PCGroup([6,-2,2,-2,2,-2,-2,48,73,362,476,86,489,1444,730,88]);
 
// Polycyclic
 
G:=Group<a,b,c|a^8=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^3,c*b*c^-1=a^5*b>;
 
// generators/relations
 

Export

Subgroup lattice of D8⋊2C4 in TeX
Character table of D8⋊2C4 in TeX

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