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G = C4.10D8  order 64 = 26

2nd non-split extension by C4 of D8 acting via D8/D4=C2

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

Aliases: C4.10D8, C4.5Q16, C4.6SD16, C42.4C22, C4⋊C4.2C4, C4⋊C8.2C2, C4⋊Q8.1C2, (C2×C4).109D4, C2.5(D4⋊C4), C2.4(Q8⋊C4), C2.4(C4.10D4), C22.40(C22⋊C4), (C2×C4).13(C2×C4), SmallGroup(64,13)

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

Derived series
C1 — C2×C4 — C4.10D8
Chief series
C1 — C2 — C22 — C2×C4 — C42 — C4⋊Q8 — C4.10D8
Lower central
C1 — C22 — C2×C4 — C4.10D8
Upper central
C1 — C22 — C42 — C4.10D8
Jennings
C1 — C22 — C22 — C42 — C4.10D8

Generators and relations for C4.10D8
 G = < a,b,c | a4=b8=1, c2=bab-1=a-1, ac=ca, cbc-1=ab-1 >

C1
C2
C2
C2
C4
C4
C22
C4
C4
2C4
4C4
4C4
C2×C4
C2×C4
C2×C4
2C2×C4
2C2×C4
4C8
4Q8
4C8
4Q8
C4⋊C4
C42
C4⋊C4
2C2×C8
2C2×C8
2C2×Q8
2C4⋊C4
C4⋊C8
C4⋊C8
C4⋊Q8
C4.10D8

Character table of C4.10D8

 class 12A2B2C4A4B4C4D4E4F4G8A8B8C8D8E8F8G8H
 size 1111222248844444444
ρ11111111111111111111    trivial
ρ211111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ3111111111-1-11-1-111-1-11    linear of order 2
ρ4111111111-1-1-111-1-111-1    linear of order 2
ρ51111-1-1-1-11-11-i-iii-ii-ii    linear of order 4
ρ61111-1-1-1-11-11ii-i-ii-ii-i    linear of order 4
ρ71111-1-1-1-111-1-ii-ii-i-iii    linear of order 4
ρ81111-1-1-1-111-1i-ii-iii-i-i    linear of order 4
ρ92222-222-2-20000000000    orthogonal lifted from D4
ρ1022222-2-22-20000000000    orthogonal lifted from D4
ρ112-22-2200-20000-√2-√200√2√20    orthogonal lifted from D8
ρ122-22-2200-20000√2√200-√2-√20    orthogonal lifted from D8
ρ132-2-220-220000-√200√2√200-√2    symplectic lifted from Q16, Schur index 2
ρ142-2-220-220000√200-√2-√200√2    symplectic lifted from Q16, Schur index 2
ρ152-22-2-20020000-√-2√-200-√-2√-20    complex lifted from SD16
ρ162-22-2-20020000√-2-√-200√-2-√-20    complex lifted from SD16
ρ172-2-2202-20000√-200√-2-√-200-√-2    complex lifted from SD16
ρ182-2-2202-20000-√-200-√-2√-200√-2    complex lifted from SD16
ρ1944-4-4000000000000000    symplectic lifted from C4.10D4, Schur index 2

Smallest permutation representation of C4.10D8
►Regular action on 64 points
Generators in S64
(1 25 11 45)(2 46 12 26)(3 27 13 47)(4 48 14 28)(5 29 15 41)(6 42 16 30)(7 31 9 43)(8 44 10 32)(17 35 61 55)(18 56 62 36)(19 37 63 49)(20 50 64 38)(21 39 57 51)(22 52 58 40)(23 33 59 53)(24 54 60 34)

(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)(33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)

(1 52 45 22 11 40 25 58)(2 21 26 51 12 57 46 39)(3 50 47 20 13 38 27 64)(4 19 28 49 14 63 48 37)(5 56 41 18 15 36 29 62)(6 17 30 55 16 61 42 35)(7 54 43 24 9 34 31 60)(8 23 32 53 10 59 44 33)
 
G:=sub<Sym(64)| (1,25,11,45)(2,46,12,26)(3,27,13,47)(4,48,14,28)(5,29,15,41)(6,42,16,30)(7,31,9,43)(8,44,10,32)(17,35,61,55)(18,56,62,36)(19,37,63,49)(20,50,64,38)(21,39,57,51)(22,52,58,40)(23,33,59,53)(24,54,60,34), (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)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,52,45,22,11,40,25,58)(2,21,26,51,12,57,46,39)(3,50,47,20,13,38,27,64)(4,19,28,49,14,63,48,37)(5,56,41,18,15,36,29,62)(6,17,30,55,16,61,42,35)(7,54,43,24,9,34,31,60)(8,23,32,53,10,59,44,33)>;
 
G:=Group( (1,25,11,45)(2,46,12,26)(3,27,13,47)(4,48,14,28)(5,29,15,41)(6,42,16,30)(7,31,9,43)(8,44,10,32)(17,35,61,55)(18,56,62,36)(19,37,63,49)(20,50,64,38)(21,39,57,51)(22,52,58,40)(23,33,59,53)(24,54,60,34), (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)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,52,45,22,11,40,25,58)(2,21,26,51,12,57,46,39)(3,50,47,20,13,38,27,64)(4,19,28,49,14,63,48,37)(5,56,41,18,15,36,29,62)(6,17,30,55,16,61,42,35)(7,54,43,24,9,34,31,60)(8,23,32,53,10,59,44,33) );
 
G=PermutationGroup([[(1,25,11,45),(2,46,12,26),(3,27,13,47),(4,48,14,28),(5,29,15,41),(6,42,16,30),(7,31,9,43),(8,44,10,32),(17,35,61,55),(18,56,62,36),(19,37,63,49),(20,50,64,38),(21,39,57,51),(22,52,58,40),(23,33,59,53),(24,54,60,34)], [(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),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64)], [(1,52,45,22,11,40,25,58),(2,21,26,51,12,57,46,39),(3,50,47,20,13,38,27,64),(4,19,28,49,14,63,48,37),(5,56,41,18,15,36,29,62),(6,17,30,55,16,61,42,35),(7,54,43,24,9,34,31,60),(8,23,32,53,10,59,44,33)]])
 

C4.10D8 is a maximal subgroup of
 C42.409D4  C42.410D4  C42.412D4  C42.414D4  C42.78D4  C42.416D4  C42.79D4  C42.81D4  C42.417D4  C42.418D4  C42.83D4  C42.84D4  C42.86D4  C42.88D4  D4⋊3D8  Q8⋊6SD16  Q8⋊3D8  C42.189C23  D4.3Q16  Q8⋊4Q16  D4⋊4Q16  Q8⋊4SD16  D4⋊4SD16  Dic5.D8
 C4p.D8: C8.28D8  C8.D8  C12.47D8  C12.2D8  C12.10D8  C20.47D8  C20.2D8  C20.10D8 ...
 C2.(D4.pD4): D4.SD16  Q8.Q16  C42.199C23  D4.7D8  C42.211C23  C42.213C23  Q8.SD16  C8⋊8D8 ...
C4.10D8 is a maximal quotient of
 C4⋊C4⋊C8  C42.8Q8  Dic5.D8
 C4.D8p: C4.10D16  C12.2D8  C20.2D8  C28.2D8 ...
 C4p.Q16: C8.16Q16  C4.6Q32  C12.47D8  C12.10D8  C20.47D8  C20.10D8  C28.47D8  C28.10D8 ...

Matrix representation of C4.10D8 ►in GL4(𝔽17) generated by

1000
0100
00162
00161
,
3300
14300
00162
0001
,
41100
111300
0007
0057
G:=sub<GL(4,GF(17))| [1,0,0,0,0,1,0,0,0,0,16,16,0,0,2,1],[3,14,0,0,3,3,0,0,0,0,16,0,0,0,2,1],[4,11,0,0,11,13,0,0,0,0,0,5,0,0,7,7] >;
 

C4.10D8 in GAP, Magma, Sage, TeX 

C_4._{10}D_8
 
% in TeX
 
G:=Group("C4.10D8");
 
// GroupNames label
 
G:=SmallGroup(64,13);
 
// by ID
 
G=gap.SmallGroup(64,13);
 
# by ID
 
G:=PCGroup([6,-2,2,-2,2,-2,2,48,73,103,362,332,158,681,165]);
 
// Polycyclic
 
G:=Group<a,b,c|a^4=b^8=1,c^2=b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=a*b^-1>;
 
// generators/relations
 

Export

Subgroup lattice of C4.10D8 in TeX
Character table of C4.10D8 in TeX

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