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G = C4⋊D8  order 64 = 26

The semidirect product of C4 and D8 acting via D8/D4=C2

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

Aliases: C4⋊2D8, D4⋊1D4, C42.16C22, C4⋊C8⋊3C2, (C4×D4)⋊3C2, (C2×D8)⋊3C2, C2.5(C2×D8), C4⋊1D4⋊2C2, (C2×C4).26D4, C4.29(C2×D4), D4⋊C4⋊6C2, (C2×C8).3C22, C4.39(C4○D4), C4⋊C4.56C22, C2.9(C8⋊C22), (C2×C4).87C23, C22.83(C2×D4), C2.11(C4⋊D4), (C2×D4).11C22, SmallGroup(64,140)

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

Derived series
C1 — C2×C4 — C4⋊D8
Chief series
C1 — C2 — C4 — C2×C4 — C2×D4 — C4×D4 — C4⋊D8
Lower central
C1 — C2 — C2×C4 — C4⋊D8
Upper central
C1 — C22 — C42 — C4⋊D8
Jennings
C1 — C2 — C2 — C2×C4 — C4⋊D8

Generators and relations for C4⋊D8
 G = < a,b,c | a4=b8=c2=1, bab-1=cac=a-1, cbc=b-1 >

Subgroups: 157 in 70 conjugacy classes, 29 normal (17 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, D4, C23, C42, C22⋊C4, C4⋊C4, C2×C8, D8, C22×C4, C2×D4, C2×D4, C2×D4, D4⋊C4, C4⋊C8, C4×D4, C4⋊1D4, C2×D8, C4⋊D8
Quotients: C1, C2, C22, D4, C23, D8, C2×D4, C4○D4, C4⋊D4, C2×D8, C8⋊C22, C4⋊D8

Character table of C4⋊D8

 class 12A2B2C2D2E2F2G4A4B4C4D4E4F4G8A8B8C8D
 size 1111448822224444444
ρ11111111111111111111    trivial
ρ21111-1-1-1111-1-111-1-11-11    linear of order 2
ρ31111111-111-1-1-1-1-1-11-11    linear of order 2
ρ41111-1-1-1-11111-1-111111    linear of order 2
ρ51111-1-11-111-1-111-11-11-1    linear of order 2
ρ6111111-1-11111111-1-1-1-1    linear of order 2
ρ7111111-1111-1-1-1-1-11-11-1    linear of order 2
ρ81111-1-1111111-1-11-1-1-1-1    linear of order 2
ρ922220000-2-22200-20000    orthogonal lifted from D4
ρ1022220000-2-2-2-20020000    orthogonal lifted from D4
ρ112-22-2-22002-2000000000    orthogonal lifted from D4
ρ122-22-22-2002-2000000000    orthogonal lifted from D4
ρ132-2-220000002-2000-√2√2√2-√2    orthogonal lifted from D8
ρ142-2-220000002-2000√2-√2-√2√2    orthogonal lifted from D8
ρ152-2-22000000-22000-√2-√2√2√2    orthogonal lifted from D8
ρ162-2-22000000-22000√2√2-√2-√2    orthogonal lifted from D8
ρ172-22-20000-22002i-2i00000    complex lifted from C4○D4
ρ182-22-20000-2200-2i2i00000    complex lifted from C4○D4
ρ1944-4-4000000000000000    orthogonal lifted from C8⋊C22

Smallest permutation representation of C4⋊D8
►On 32 points
Generators in S32
(1 18 30 10)(2 11 31 19)(3 20 32 12)(4 13 25 21)(5 22 26 14)(6 15 27 23)(7 24 28 16)(8 9 29 17)

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

C4⋊D8 is a maximal subgroup of
 D4⋊D8  Q8⋊D8  D4⋊3D8  Q8⋊3D8  C42.189C23  D4.2SD16  D4.2D8  C42.248C23  C42.252C23  C42.443D4  C42.444D4  C42.18C23  C42.221D4  C42.450D4  C42.227D4  C42.233D4  C42.353C23  C42.356C23  C42.263D4  C42.270D4  Q8⋊4D8  C42.502C23  C42.507C23  C42.511C23  C42.514C23  Q8⋊5D8
 D4p⋊D4: D8⋊9D4  D8⋊4D4  D12⋊3D4  C4⋊D24  C12⋊2D8  D20⋊3D4  C4⋊D40  C20⋊2D8 ...
 C4p⋊D8: C8⋊8D8  C8⋊7D8  C8⋊D8  C8⋊2D8  C12⋊7D8  C20⋊7D8  C28⋊7D8 ...
 (Cp×D4)⋊D4: C42.211D4  C42.14C23  Dic3⋊D8  Dic5⋊D8  Dic7⋊D8 ...
 C8⋊pD4⋊C2: D4⋊SD16  C42.185C23  C42.272D4  C42.275D4  C42.406C23  C42.410C23  C42.293D4  C42.295D4 ...
C4⋊D8 is a maximal quotient of
 C8.28D8  C8.D8  C42.98D4  (C2×C4)⋊9D8  D4⋊C4⋊C4  C42.29Q8  C42.118D4  C4⋊C4⋊7D4  (C2×C4)⋊3D8  (C2×C4).23D8  (C2×C4).24D8  (C2×C8).1Q8  Q16⋊2D4  D8.4D4  Q16.4D4  D8.5D4  Q16.5D4  C8.3D8  C8.5D8
 D4p⋊D4: D8⋊2D4  D8⋊3D4  D12⋊3D4  C4⋊D24  C12⋊2D8  D20⋊3D4  C4⋊D40  C20⋊2D8 ...
 C4p⋊D8: C8⋊8D8  C8⋊7D8  C8⋊D8  C8⋊2D8  C12⋊7D8  C20⋊7D8  C28⋊7D8 ...
 (Cp×D4)⋊D4: (C2×C4)⋊2D8  Dic3⋊D8  Dic5⋊D8  Dic7⋊D8 ...

Matrix representation of C4⋊D8 ►in GL4(𝔽17) generated by

13900
0400
00160
00016
,
13000
4400
00314
0033
,
13900
4400
00160
0001
G:=sub<GL(4,GF(17))| [13,0,0,0,9,4,0,0,0,0,16,0,0,0,0,16],[13,4,0,0,0,4,0,0,0,0,3,3,0,0,14,3],[13,4,0,0,9,4,0,0,0,0,16,0,0,0,0,1] >;
 

C4⋊D8 in GAP, Magma, Sage, TeX 

C_4\rtimes D_8
 
% in TeX
 
G:=Group("C4:D8");
 
// GroupNames label
 
G:=SmallGroup(64,140);
 
// by ID
 
G=gap.SmallGroup(64,140);
 
# by ID
 
G:=PCGroup([6,-2,2,2,-2,2,-2,121,55,362,1444,376,88]);
 
// Polycyclic
 
G:=Group<a,b,c|a^4=b^8=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;
 
// generators/relations
 

Export

Character table of C4⋊D8 in TeX

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