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G = C4×D8  order 64 = 26

Direct product of C4 and D8

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

Aliases: C4×D8, C42.71C22, C8⋊4(C2×C4), (C4×C8)⋊7C2, (C4×D4)⋊1C2, D4⋊1(C2×C4), C2.3(C2×D8), C4○2(C2.D8), C2.D8⋊14C2, (C2×D8).7C2, (C2×C4).51D4, C2.12(C4×D4), C4.1(C4○D4), C2.3(C4○D8), C4.9(C22×C4), C4○2(D4⋊C4), D4⋊C4⋊21C2, C4⋊C4.50C22, (C2×C4).73C23, (C2×C8).75C22, C22.51(C2×D4), (C2×D4).50C22, SmallGroup(64,118)

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

Derived series
C1 — C4 — C4×D8
Chief series
C1 — C2 — C22 — C2×C4 — C42 — C4×D4 — C4×D8
Lower central
C1 — C2 — C4 — C4×D8
Upper central
C1 — C2×C4 — 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, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 125 in 67 conjugacy classes, 37 normal (17 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, D4, D4, C23, C42, C22⋊C4, C4⋊C4, C2×C8, D8, C22×C4, C2×D4, C4×C8, D4⋊C4, C2.D8, C4×D4, C2×D8, C4×D8
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D8, C22×C4, C2×D4, C4○D4, C4×D4, C2×D8, C4○D8, C4×D8

Character table of C4×D8

 class 12A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J4K4L8A8B8C8D8E8F8G8H
 size 1111444411112222444422222222
ρ11111111111111111111111111111    trivial
ρ211111-11-111111111-111-1-1-1-1-1-1-1-1-1    linear of order 2
ρ31111-11-11111111111-1-11-1-1-1-1-1-1-1-1    linear of order 2
ρ41111-1-1-1-111111111-1-1-1-111111111    linear of order 2
ρ511111111-1-1-1-1-1-111-1-1-1-11-1-1111-1-1    linear of order 2
ρ61111-1-1-1-1-1-1-1-1-1-11111111-1-1111-1-1    linear of order 2
ρ711111-11-1-1-1-1-1-1-1111-1-11-111-1-1-111    linear of order 2
ρ81111-11-11-1-1-1-1-1-111-111-1-111-1-1-111    linear of order 2
ρ91-1-1111-1-1ii-i-i-ii-11-ii-ii-1-ii11-1-ii    linear of order 4
ρ101-1-1111-1-1-i-iiii-i-11i-ii-i-1i-i11-1i-i    linear of order 4
ρ111-1-11-111-1ii-i-i-ii-11-i-iii1i-i-1-11i-i    linear of order 4
ρ121-1-11-111-1-i-iiii-i-11ii-i-i1-ii-1-11-ii    linear of order 4
ρ131-1-11-1-111ii-i-i-ii-11i-ii-i-1-ii11-1-ii    linear of order 4
ρ141-1-11-1-111-i-iiii-i-11-ii-ii-1i-i11-1i-i    linear of order 4
ρ151-1-111-1-11ii-i-i-ii-11ii-i-i1i-i-1-11i-i    linear of order 4
ρ161-1-111-1-11-i-iiii-i-11-i-iii1-ii-1-11-ii    linear of order 4
ρ17222200002222-2-2-2-2000000000000    orthogonal lifted from D4
ρ1822220000-2-2-2-222-2-2000000000000    orthogonal lifted from D4
ρ1922-2-200002-2-2200000000-√2-√2√2-√2√2√2√2-√2    orthogonal lifted from D8
ρ2022-2-20000-222-200000000-√2√2-√2-√2√2√2-√2√2    orthogonal lifted from D8
ρ2122-2-200002-2-2200000000√2√2-√2√2-√2-√2-√2√2    orthogonal lifted from D8
ρ2222-2-20000-222-200000000√2-√2√2√2-√2-√2√2-√2    orthogonal lifted from D8
ρ232-22-200002i-2i2i-2i00000000-√2-√-2-√-2√2-√2√2√-2√-2    complex lifted from C4○D8
ρ242-2-220000-2i-2i2i2i-2i2i2-2000000000000    complex lifted from C4○D4
ρ252-22-20000-2i2i-2i2i00000000-√2√-2√-2√2-√2√2-√-2-√-2    complex lifted from C4○D8
ρ262-22-20000-2i2i-2i2i00000000√2-√-2-√-2-√2√2-√2√-2√-2    complex lifted from C4○D8
ρ272-22-200002i-2i2i-2i00000000√2√-2√-2-√2√2-√2-√-2-√-2    complex lifted from C4○D8
ρ282-2-2200002i2i-2i-2i2i-2i2-2000000000000    complex lifted from C4○D4

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

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

C4×D8 is a maximal subgroup of
 C4.16D16  D8⋊C8  D8⋊5C8  C8⋊9D8  D4⋊2M4(2)  C8⋊6D8  C8⋊3M4(2)  SD32⋊3C4  D8⋊2D4  D8.4D4  D8.5D4  D8⋊1Q8  D8⋊Q8  D8.Q8  C42.221D4  C42.384D4  C42.225D4  C42.450D4  C42.352C23  C42.353C23  C42.356C23  C42.358C23  C42.308D4  C42.387C23  D8⋊5D4  D8⋊13D4  C42.468C23  C42.479C23  Q8⋊4D8  C42.501C23  C42.502C23  C42.507C23  C42.508C23  C42.511C23  D8⋊6Q8  D8⋊4Q8  D8⋊5Q8  Q8⋊5D8  C42.527C23  C42.530C23
 D8p⋊C4: D16⋊4C4  Dic3⋊5D8  D40⋊12C4  Dic7⋊5D8 ...
 C8⋊pD4⋊C2: C42.366D4  C42.255D4  C42.388C23  C42.391C23  D8⋊4D4  D8⋊12D4  D4⋊4D8  C42.462C23 ...
 D4p⋊(C2×C4): C42.275C23  C42.277C23  C42.280C23  Dic3⋊4D8  Dic5⋊4D8  Dic7⋊4D8 ...
C4×D8 is a maximal quotient of
 C8⋊9D8  C8⋊6D8  C2.D8⋊4C4  C8⋊5(C4⋊C4)
 D8p⋊C4: D16⋊4C4  C8○D16  D16⋊5C4  Dic3⋊5D8  D40⋊12C4  Dic7⋊5D8 ...
 C2p.(C4×D4): C2.(C4×D8)  (C2×C4)⋊9D8  D4⋊C4⋊C4  C2.(C8⋊7D4)  (C2×C4)⋊6D8  SD32⋊3C4  Q32⋊4C4  Dic3⋊4D8 ...

Matrix representation of C4×D8 ►in GL3(𝔽17) generated by

400
0160
0016
,
1600
0143
01414
,
100
010
0016
G:=sub<GL(3,GF(17))| [4,0,0,0,16,0,0,0,16],[16,0,0,0,14,14,0,3,14],[1,0,0,0,1,0,0,0,16] >;
 

C4×D8 in GAP, Magma, Sage, TeX 

C_4\times D_8
 
% in TeX
 
G:=Group("C4xD8");
 
// GroupNames label
 
G:=SmallGroup(64,118);
 
// by ID
 
G=gap.SmallGroup(64,118);
 
# by ID
 
G:=PCGroup([6,-2,2,2,-2,2,-2,96,121,86,963,489,117]);
 
// Polycyclic
 
G:=Group<a,b,c|a^4=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
 
// generators/relations
 

Export

Character table of C4×D8 in TeX

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