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G = C22×D8  order 64 = 26

Direct product of C22 and D8

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

Aliases: C22×D8, C8⋊2C23, D4⋊1C23, C4.1C24, C23.60D4, (C22×C8)⋊7C2, C4.16(C2×D4), (C2×C4).87D4, (C2×C8)⋊12C22, (C22×D4)⋊10C2, (C2×D4)⋊14C22, C2.23(C22×D4), C22.64(C2×D4), (C2×C4).135C23, (C22×C4).129C22, 2-Sylow(GO-(4,7)), SmallGroup(64,250)

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

Derived series
C1 — C4 — C22×D8
Chief series
C1 — C2 — C4 — C2×C4 — C22×C4 — C22×D4 — C22×D8
Lower central
C1 — C2 — C4 — C22×D8
Upper central
C1 — C23 — C22×C4 — C22×D8
Jennings
C1 — C2 — C2 — C4 — C22×D8

Generators and relations for C22×D8
 G = < a,b,c,d | a2=b2=c8=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 329 in 169 conjugacy classes, 89 normal (7 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C8, C2×C4, D4, D4, C23, C23, C2×C8, D8, C22×C4, C2×D4, C2×D4, C24, C22×C8, C2×D8, C22×D4, C22×D8
Quotients: C1, C2, C22, D4, C23, D8, C2×D4, C24, C2×D8, C22×D4, C22×D8

Character table of C22×D8

 class 12A2B2C2D2E2F2G2H2I2J2K2L2M2N2O4A4B4C4D8A8B8C8D8E8F8G8H
 size 1111111144444444222222222222
ρ11111111111111111111111111111    trivial
ρ211-111-1-1-11-1-111-1-111-1-111-1-111-1-11    linear of order 2
ρ31-11-11-11-1-11-111-11-11-11-111-11-11-1-1    linear of order 2
ρ41-1-1-111-11-1-11111-1-111-1-11-111-1-11-1    linear of order 2
ρ51-1-1-111-1111-1-1-1-11111-1-11-111-1-11-1    linear of order 2
ρ611111111-1-1-1-1-1-1-1-1111111111111    linear of order 2
ρ71-11-11-11-11-11-1-11-111-11-111-11-11-1-1    linear of order 2
ρ811-111-1-1-1-111-1-111-11-1-111-1-111-1-11    linear of order 2
ρ911111111-1-1-1-111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ101-1-1-111-1111-1-111-1-111-1-1-11-1-111-11    linear of order 2
ρ1111-111-1-1-1-111-11-1-111-1-11-111-1-111-1    linear of order 2
ρ121-11-11-11-11-11-11-11-11-11-1-1-11-11-111    linear of order 2
ρ131-11-11-11-1-11-11-11-111-11-1-1-11-11-111    linear of order 2
ρ1411-111-1-1-11-1-11-111-11-1-11-111-1-111-1    linear of order 2
ρ151-1-1-111-11-1-111-1-11111-1-1-11-1-111-11    linear of order 2
ρ16111111111111-1-1-1-11111-1-1-1-1-1-1-1-1    linear of order 2
ρ172-2-2-222-2200000000-2-22200000000    orthogonal lifted from D4
ρ182-22-22-22-200000000-22-2200000000    orthogonal lifted from D4
ρ1922-222-2-2-200000000-222-200000000    orthogonal lifted from D4
ρ202222222200000000-2-2-2-200000000    orthogonal lifted from D4
ρ212-222-2-2-22000000000000-√2√2√2√2-√2-√2-√2√2    orthogonal lifted from D8
ρ222-2-22-222-2000000000000√2√2√2-√2√2-√2-√2-√2    orthogonal lifted from D8
ρ232-2-22-222-2000000000000-√2-√2-√2√2-√2√2√2√2    orthogonal lifted from D8
ρ242-222-2-2-22000000000000√2-√2-√2-√2√2√2√2-√2    orthogonal lifted from D8
ρ25222-2-22-2-2000000000000-√2√2-√2√2√2-√2√2-√2    orthogonal lifted from D8
ρ2622-2-2-2-222000000000000√2√2-√2-√2-√2-√2√2√2    orthogonal lifted from D8
ρ2722-2-2-2-222000000000000-√2-√2√2√2√2√2-√2-√2    orthogonal lifted from D8
ρ28222-2-22-2-2000000000000√2-√2√2-√2-√2√2-√2√2    orthogonal lifted from D8

Smallest permutation representation of C22×D8
►On 32 points
Generators in S32
(1 16)(2 9)(3 10)(4 11)(5 12)(6 13)(7 14)(8 15)(17 30)(18 31)(19 32)(20 25)(21 26)(22 27)(23 28)(24 29)

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

C22×D8 is a maximal subgroup of
 (C2×C4)⋊9D8  (C2×C4)⋊6D8  (C2×D8)⋊10C4  M4(2).32D4  C23⋊2D8  (C2×C4)⋊2D8  (C22×D8).C2  M4(2).4D4  C4⋊C4.96D4  (C2×C4)⋊3D8  C23.40D8  D8⋊7D4  D8.9D4  C42.277C23  C4○D4⋊D4  C42.14C23  (C2×C8)⋊12D4  M4(2)⋊11D4  D8⋊9D4  D8⋊12D4
C22×D8 is a maximal quotient of
 C42.221D4  C42.366D4  C23⋊3D8  C42.263D4  C42.278D4  C42.293D4  D4⋊4D8  D4⋊5D8  Q8⋊4D8  Q8⋊5D8  D16⋊C22  D4○D16  D4○SD32  Q8○D16

Matrix representation of C22×D8 ►in GL4(𝔽17) generated by

16000
0100
0010
0001
,
16000
01600
0010
0001
,
1000
01600
00314
0033
,
1000
0100
0010
00016
G:=sub<GL(4,GF(17))| [16,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[16,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,16,0,0,0,0,3,3,0,0,14,3],[1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,16] >;
 

C22×D8 in GAP, Magma, Sage, TeX 

C_2^2\times D_8
 
% in TeX
 
G:=Group("C2^2xD8");
 
// GroupNames label
 
G:=SmallGroup(64,250);
 
// by ID
 
G=gap.SmallGroup(64,250);
 
# by ID
 
G:=PCGroup([6,-2,2,2,2,-2,-2,217,1444,730,88]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^2=b^2=c^8=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
 
// generators/relations
 

Export

Character table of C22×D8 in TeX

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