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G = C8⋊5D4  order 64 = 26

2nd semidirect product of C8 and D4 acting via D4/C4=C2

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

Aliases: C8⋊5D4, C4⋊1SD16, C42.79C22, C4⋊Q8⋊7C2, (C4×C8)⋊12C2, C4.1(C2×D4), (C2×C4).76D4, C4⋊1D4.6C2, (C2×SD16)⋊14C2, C2.5(C4⋊1D4), (C2×C8).92C22, C2.16(C2×SD16), (C2×C4).117C23, (C2×D4).28C22, C22.113(C2×D4), (C2×Q8).24C22, SmallGroup(64,173)

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

Derived series
C1 — C2×C4 — C8⋊5D4
Chief series
C1 — C2 — C22 — C2×C4 — C42 — C4×C8 — C8⋊5D4
Lower central
C1 — C2 — C2×C4 — C8⋊5D4
Upper central
C1 — C22 — C42 — C8⋊5D4
Jennings
C1 — C2 — C2 — C2×C4 — C8⋊5D4

Generators and relations for C8⋊5D4
 G = < a,b,c | a8=b4=c2=1, ab=ba, cac=a3, cbc=b-1 >

Subgroups: 145 in 71 conjugacy classes, 33 normal (9 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C42, C4⋊C4, C2×C8, SD16, C2×D4, C2×D4, C2×Q8, C4×C8, C4⋊1D4, C4⋊Q8, C2×SD16, C8⋊5D4
Quotients: C1, C2, C22, D4, C23, SD16, C2×D4, C4⋊1D4, C2×SD16, C8⋊5D4

Character table of C8⋊5D4

 class 12A2B2C2D2E4A4B4C4D4E4F4G4H8A8B8C8D8E8F8G8H
 size 1111882222228822222222
ρ11111111111111111111111    trivial
ρ211111-1-1-11-11-1-11-11-111-1-11    linear of order 2
ρ3111111111111-1-1-1-1-1-1-1-1-1-1    linear of order 2
ρ411111-1-1-11-11-11-11-11-1-111-1    linear of order 2
ρ51111-1-1111111-1-111111111    linear of order 2
ρ61111-11-1-11-11-11-1-11-111-1-11    linear of order 2
ρ71111-1-111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ81111-11-1-11-11-1-111-11-1-111-1    linear of order 2
ρ92-2-22000020-2000-202002-20    orthogonal lifted from D4
ρ102222002-2-2-2-220000000000    orthogonal lifted from D4
ρ112-2-22000020-200020-200-220    orthogonal lifted from D4
ρ122-2-220000-2020000202-200-2    orthogonal lifted from D4
ρ132-2-220000-2020000-20-22002    orthogonal lifted from D4
ρ14222200-22-22-2-20000000000    orthogonal lifted from D4
ρ1522-2-20020000-200-√-2-√-2-√-2√-2√-2√-2√-2-√-2    complex lifted from SD16
ρ1622-2-20020000-200√-2√-2√-2-√-2-√-2-√-2-√-2√-2    complex lifted from SD16
ρ1722-2-200-20000200-√-2√-2-√-2-√-2-√-2√-2√-2√-2    complex lifted from SD16
ρ182-22-2000-2020000-√-2√-2√-2-√-2√-2-√-2√-2-√-2    complex lifted from SD16
ρ192-22-200020-20000-√-2-√-2√-2√-2-√-2-√-2√-2√-2    complex lifted from SD16
ρ2022-2-200-20000200√-2-√-2√-2√-2√-2-√-2-√-2-√-2    complex lifted from SD16
ρ212-22-200020-20000√-2√-2-√-2-√-2√-2√-2-√-2-√-2    complex lifted from SD16
ρ222-22-2000-2020000√-2-√-2-√-2√-2-√-2√-2-√-2√-2    complex lifted from SD16

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

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

C8⋊5D4 is a maximal subgroup of
 C8.29D8  (C4×C8)⋊6C4  (C4×C8).C4  D4⋊SD16  Q8⋊SD16  C42.185C23  D4⋊2SD16  C42.191C23  Q8⋊2SD16  C8⋊8D8  C8⋊D8  C82⋊12C2  C42.664C23  C42.666C23  C8⋊3D8  C8.2D8  M4(2)⋊7D4  C42.365D4  C42.386C23  C42.259D4  C42.264D4  C42.265D4  C42.266D4  C42.268D4  C42.407C23  C42.408C23  D8⋊4D4  D4×SD16  Q8⋊9SD16  C42.528C23  C42.72C23  C42.75C23
 C4p⋊SD16: C8⋊14SD16  C8⋊SD16  C8⋊8SD16  C8⋊5D8  C8⋊5SD16  C8⋊6SD16  C8⋊5D12  C12⋊4SD16 ...
 C4p.(C2×D4): C42.360D4  M4(2)⋊8D4  M4(2)⋊10D4  Q16⋊5D4  SD16⋊11D4  D8⋊6D4  C24⋊15D4  C40⋊15D4 ...
C8⋊5D4 is a maximal quotient of
 C8⋊5Q16  C82⋊12C2  C8.9SD16  C42.58Q8  C42.431D4  C42.432D4  (C2×C4)⋊9SD16  (C2×C8).169D4  (C2×C8).170D4
 C4p⋊SD16: C8⋊8SD16  C8⋊5D8  C8⋊5SD16  C8⋊6SD16  C8⋊5D12  C12⋊4SD16  C12⋊6SD16  C8⋊5D20 ...
 (C2×D4).D2p: (C2×C4)⋊3SD16  (C2×C8)⋊20D4  C24⋊15D4  C40⋊15D4  C56⋊15D4 ...

Matrix representation of C8⋊5D4 ►in GL4(𝔽17) generated by

12500
121200
00160
00016
,
16000
01600
0001
00160
,
16000
0100
0001
0010
G:=sub<GL(4,GF(17))| [12,12,0,0,5,12,0,0,0,0,16,0,0,0,0,16],[16,0,0,0,0,16,0,0,0,0,0,16,0,0,1,0],[16,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0] >;
 

C8⋊5D4 in GAP, Magma, Sage, TeX 

C_8\rtimes_5D_4
 
% in TeX
 
G:=Group("C8:5D4");
 
// GroupNames label
 
G:=SmallGroup(64,173);
 
// by ID
 
G=gap.SmallGroup(64,173);
 
# by ID
 
G:=PCGroup([6,-2,2,2,-2,2,-2,121,55,362,86,963,117]);
 
// Polycyclic
 
G:=Group<a,b,c|a^8=b^4=c^2=1,a*b=b*a,c*a*c=a^3,c*b*c=b^-1>;
 
// generators/relations
 

Export

Character table of C8⋊5D4 in TeX

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