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G = C4.10D8order 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

C1C2×C4 — C4.10D8
C1C2C22C2×C4C42C4⋊Q8 — C4.10D8
C1C22C2×C4 — C4.10D8
C1C22C42 — C4.10D8
C1C22C22C42 — 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 >

2C4
4C4
4C4
2C2×C4
2C2×C4
4C8
4Q8
4C8
4Q8
2C2×C8
2C2×C8
2C2×Q8
2C4⋊C4

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
ρ112-22-2200-20000-2-200220    orthogonal lifted from D8
ρ122-22-2200-200002200-2-20    orthogonal lifted from D8
ρ132-2-220-220000-2002200-2    symplectic lifted from Q16, Schur index 2
ρ142-2-220-220000200-2-2002    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  D43D8  Q86SD16  Q83D8  C42.189C23  D4.3Q16  Q84Q16  D44Q16  Q84SD16  D44SD16  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  C88D8 ...
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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