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G = C4.4D8  order 64 = 26

4th non-split extension by C4 of D8 acting via D8/C8=C2

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

Aliases: C4.4D8, C4.3SD16, C42.76C22, (C4×C8)⋊5C2, C4⋊Q8⋊5C2, C2.9(C2×D8), (C2×C4).74D4, D4⋊C4⋊3C2, C4⋊1D4.4C2, C4.13(C4○D4), C4⋊C4.16C22, (C2×C8).66C22, C2.14(C2×SD16), (C2×C4).111C23, C2.9(C4.4D4), (C2×D4).24C22, C22.107(C2×D4), SmallGroup(64,167)

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

Derived series
C1 — C2×C4 — C4.4D8
Chief series
C1 — C2 — C4 — C2×C4 — C2×C8 — C4×C8 — C4.4D8
Lower central
C1 — C2 — C2×C4 — C4.4D8
Upper central
C1 — C22 — C42 — C4.4D8
Jennings
C1 — C2 — C2 — C2×C4 — C4.4D8

Generators and relations for C4.4D8
 G = < a,b,c | a4=b8=1, c2=a2, ab=ba, cac-1=a-1, cbc-1=a2b-1 >

Subgroups: 129 in 59 conjugacy classes, 29 normal (13 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, Q8, C23, C42, C4⋊C4, C4⋊C4, C2×C8, C2×D4, C2×D4, C2×Q8, C4×C8, D4⋊C4, C4⋊1D4, C4⋊Q8, C4.4D8
Quotients: C1, C2, C22, D4, C23, D8, SD16, C2×D4, C4○D4, C4.4D4, C2×D8, C2×SD16, C4.4D8

Character table of C4.4D8

 class 12A2B2C2D2E4A4B4C4D4E4F4G4H8A8B8C8D8E8F8G8H
 size 1111882222228822222222
ρ11111111111111111111111    trivial
ρ211111-1-1-11-11-11-1-11-11-11-11    linear of order 2
ρ311111-1-1-11-11-1-111-11-11-11-1    linear of order 2
ρ4111111111111-1-1-1-1-1-1-1-1-1-1    linear of order 2
ρ51111-1-1111111-1-111111111    linear of order 2
ρ61111-11-1-11-11-1-11-11-11-11-11    linear of order 2
ρ71111-11-1-11-11-11-11-11-11-11-1    linear of order 2
ρ81111-1-111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ922220022-2-2-2-20000000000    orthogonal lifted from D4
ρ10222200-2-2-22-220000000000    orthogonal lifted from D4
ρ112-2-220000020-200-√2√2-√2-√2√2-√2√2√2    orthogonal lifted from D8
ρ122-2-2200000-20200-√2-√2-√2√2√2√2√2-√2    orthogonal lifted from D8
ρ132-2-220000020-200√2-√2√2√2-√2√2-√2-√2    orthogonal lifted from D8
ρ142-2-2200000-20200√2√2√2-√2-√2-√2-√2√2    orthogonal lifted from D8
ρ152-22-20000-2020002i0-2i0-2i02i0    complex lifted from C4○D4
ρ1622-2-200-22000000-√-2-√-2√-2√-2-√-2-√-2√-2√-2    complex lifted from SD16
ρ172-22-2000020-20000-2i0-2i02i02i    complex lifted from C4○D4
ρ182-22-20000-202000-2i02i02i0-2i0    complex lifted from C4○D4
ρ1922-2-2002-2000000√-2-√-2-√-2√-2√-2-√-2-√-2√-2    complex lifted from SD16
ρ202-22-2000020-200002i02i0-2i0-2i    complex lifted from C4○D4
ρ2122-2-200-22000000√-2√-2-√-2-√-2√-2√-2-√-2-√-2    complex lifted from SD16
ρ2222-2-2002-2000000-√-2√-2√-2-√-2-√-2√-2√-2-√-2    complex lifted from SD16

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

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

C4.4D8 is a maximal subgroup of
 D4⋊3D8  Q8⋊6SD16  Q8⋊3D8  C42.189C23  D4.D8  C42.201C23  Q8.D8  Q8⋊3SD16  D4.2SD16  Q8.2SD16  C42.248C23  C42.249C23  C82⋊3C2  C42.355D4  C42.366C23  C42.240D4  C42.243D4  C42.263D4  C42.278D4  C42.279D4  C42.280D4  D4⋊7SD16  C42.469C23  C42.473C23  C42.482C23  Q8⋊4D8  C42.502C23  Q8⋊8SD16  C42.506C23  C42.507C23  C42.509C23  C42.514C23  C42.516C23
 C4p.D8: C82⋊5C2  C8.2D8  C4.5D24  C12.16D8  C12.D8  C4.5D40  C20.16D8  C20.D8 ...
 C4p⋊Q8⋊C2: C42.391C23  C42.423C23  Dic3.SD16  Dic5.5D8  Dic7.SD16 ...
 C8⋊pD4⋊C2: C8⋊5D8  C8⋊6SD16  C42.664C23  C8⋊3D8  C42.365D4  C42.366D4  C42.259D4  C42.261D4 ...
 C2.(C8⋊pD4): D4.2D8  Q8.2D8  C42.252C23  C42.253C23  C82⋊12C2  C8⋊5SD16  C42.666C23  C42.667C23 ...
C4.4D8 is a maximal quotient of
 C42.55Q8  C2.(C8⋊7D4)  C42.432D4  C42.436D4  C4⋊C4⋊7D4  (C2×C4).24D8  (C2×C4).28D8
 C4.D8p: C4.4D16  C4.5D24  C4.5D40  C4.5D56 ...
 C4p.SD16: C4.SD32  C8.22SD16  C8.12SD16  C8.13SD16  C8.14SD16  C12.16D8  C12.D8  C20.16D8 ...
 C4⋊C4.D2p: (C2×D4)⋊Q8  Dic3.SD16  Dic5.5D8  Dic7.SD16 ...

Matrix representation of C4.4D8 ►in GL4(𝔽17) generated by

16200
16100
00013
00130
,
71000
12000
0001
0010
,
71000
121000
0001
00160
G:=sub<GL(4,GF(17))| [16,16,0,0,2,1,0,0,0,0,0,13,0,0,13,0],[7,12,0,0,10,0,0,0,0,0,0,1,0,0,1,0],[7,12,0,0,10,10,0,0,0,0,0,16,0,0,1,0] >;
 

C4.4D8 in GAP, Magma, Sage, TeX 

C_4._4D_8
 
% in TeX
 
G:=Group("C4.4D8");
 
// GroupNames label
 
G:=SmallGroup(64,167);
 
// by ID
 
G=gap.SmallGroup(64,167);
 
# by ID
 
G:=PCGroup([6,-2,2,2,-2,2,-2,121,103,362,50,963,117,1444,88]);
 
// Polycyclic
 
G:=Group<a,b,c|a^4=b^8=1,c^2=a^2,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=a^2*b^-1>;
 
// generators/relations
 

Export

Character table of C4.4D8 in TeX

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