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G = D82C4order 64 = 26

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

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

Aliases: D82C4, Q162C4, C8.22D4, C4.8SD16, C22.3D8, M5(2)⋊5C2, C8.1(C2×C4), C4.Q81C2, 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

C1C8 — D82C4
C1C2C4C2×C4C2×C8C4○D8 — D82C4
C1C2C4C8 — D82C4
C1C2C2×C4C2×C8 — D82C4
C1C2C2C2C2C4C4C2×C8 — D82C4

Generators and relations for D82C4
 G = < a,b,c | a8=b2=c4=1, bab=a-1, cac-1=a3, cbc-1=a5b >

2C2
8C2
4C22
4C4
8C4
2D4
2Q8
4D4
4C2×C4
4C2×C4
2C16
2SD16
2C4○D4
2C4⋊C4

Character table of D82C4

 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
ρ112220-2-2000000-222-2    orthogonal lifted from D8
ρ122220-2-20000002-2-22    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 D82C4
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);

D82C4 is a maximal subgroup of
C23.13D8  Q32⋊C4  D8⋊D4  D8.D4  D10.D8
 D8p⋊C4: D16⋊C4  D248C4  D242C4  D4014C4  D408C4  D401C4  D568C4  D562C4 ...
 C4p.SD16: D83Q8  D8.2Q8  D82Dic3  D82Dic5  D82Dic7 ...
D82C4 is a maximal quotient of
D8⋊C8  Q16⋊C8  C22.SD32  C23.32D8  C8.C42  D10.D8  D401C4
 C8.D4p: C8.30D8  D248C4  D4014C4  D568C4 ...
 C4p.SD16: C8.16Q16  D242C4  D82Dic3  D408C4  D82Dic5  D562C4  D82Dic7 ...

Matrix representation of D82C4 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] >;

D82C4 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 D82C4 in TeX
Character table of D82C4 in TeX

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