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G = C2×C4○D4  order 32 = 25

Direct product of C2 and C4○D4

direct product, p-group, metabelian, nilpotent (class 2), monomial

Aliases: C2×C4○D4, D4⋊3C22, C2.3C24, C4.8C23, Q8⋊3C22, C22.1C23, C23.11C22, C4○(C2×D4), (C2×C4)○D4, C4○(C2×Q8), (C2×C4)○Q8, C4○(C4○D4), (C2×D4)⋊7C2, (C2×Q8)⋊6C2, (C22×C4)⋊6C2, (C2×C4)⋊5C22, (C2×C4)○(C2×Q8), SmallGroup(32,48)

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

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

Generators and relations for C2×C4○D4
 G = < a,b,c,d | a2=b4=d2=1, c2=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c >

Subgroups: 94 in 82 conjugacy classes, 70 normal (6 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C2×C4, C2×C4, D4, Q8, C23, C22×C4, C2×D4, C2×Q8, C4○D4, C2×C4○D4
Quotients: C1, C2, C22, C23, C4○D4, C24, C2×C4○D4

Character table of C2×C4○D4

 class 12A2B2C2D2E2F2G2H2I4A4B4C4D4E4F4G4H4I4J
 size 11112222221111222222
ρ111111111111111111111    trivial
ρ21111-1-11111-1-1-1-1-1-111-1-1    linear of order 2
ρ31111-1-1-1-111111111-1-1-1-1    linear of order 2
ρ4111111-1-111-1-1-1-1-1-1-1-111    linear of order 2
ρ51-11-11-11-1-1111-1-11-1-11-11    linear of order 2
ρ61-11-1-111-1-11-1-111-11-111-1    linear of order 2
ρ71-11-1-11-11-1111-1-11-11-11-1    linear of order 2
ρ81-11-11-1-11-11-1-111-111-1-11    linear of order 2
ρ911111111-1-1-1-1-1-111-1-1-1-1    linear of order 2
ρ101111-1-111-1-11111-1-1-1-111    linear of order 2
ρ111111-1-1-1-1-1-1-1-1-1-1111111    linear of order 2
ρ12111111-1-1-1-11111-1-111-1-1    linear of order 2
ρ131-11-11-11-11-1-1-1111-11-11-1    linear of order 2
ρ141-11-1-111-11-111-1-1-111-1-11    linear of order 2
ρ151-11-1-11-111-1-1-1111-1-11-11    linear of order 2
ρ161-11-11-1-111-111-1-1-11-111-1    linear of order 2
ρ1722-2-2000000-2i2i2i-2i000000    complex lifted from C4○D4
ρ182-2-22000000-2i2i-2i2i000000    complex lifted from C4○D4
ρ192-2-220000002i-2i2i-2i000000    complex lifted from C4○D4
ρ2022-2-20000002i-2i-2i2i000000    complex lifted from C4○D4

Permutation representations of C2×C4○D4
►On 16 points - transitive group 16T18
Generators in S16
(1 11)(2 12)(3 9)(4 10)(5 14)(6 15)(7 16)(8 13)

(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)

(1 4 3 2)(5 6 7 8)(9 12 11 10)(13 14 15 16)

(1 13)(2 14)(3 15)(4 16)(5 12)(6 9)(7 10)(8 11)
 
G:=sub<Sym(16)| (1,11)(2,12)(3,9)(4,10)(5,14)(6,15)(7,16)(8,13), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,4,3,2)(5,6,7,8)(9,12,11,10)(13,14,15,16), (1,13)(2,14)(3,15)(4,16)(5,12)(6,9)(7,10)(8,11)>;
 
G:=Group( (1,11)(2,12)(3,9)(4,10)(5,14)(6,15)(7,16)(8,13), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,4,3,2)(5,6,7,8)(9,12,11,10)(13,14,15,16), (1,13)(2,14)(3,15)(4,16)(5,12)(6,9)(7,10)(8,11) );
 
G=PermutationGroup([[(1,11),(2,12),(3,9),(4,10),(5,14),(6,15),(7,16),(8,13)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(1,4,3,2),(5,6,7,8),(9,12,11,10),(13,14,15,16)], [(1,13),(2,14),(3,15),(4,16),(5,12),(6,9),(7,10),(8,11)]])
 
G:=TransitiveGroup(16,18);
 

C2×C4○D4 is a maximal subgroup of
 (C22×C8)⋊C2  C23.C23  M4(2).8C22  C23.24D4  C23.36D4  C42⋊C22  D4⋊D4  D4.7D4  C23.33C23  C22.19C24  C22.26C24  C22.29C24  C23.38C23  C22.31C24  D4⋊5D4  D4⋊6D4  Q8⋊5D4  Q8⋊6D4  Q8○M4(2)  D8⋊C22  C2.C25
C2×C4○D4 is a maximal quotient of
 C2×C4×D4  C2×C4×Q8  C22.19C24  C23.36C23  C22.26C24  C23.37C23  C22.32C24  C22.33C24  C22.34C24  C22.35C24  C22.36C24  D4⋊5D4  D4⋊6D4  Q8⋊5D4  Q8⋊6D4  C22.45C24  C22.46C24  C22.47C24  D4⋊3Q8  C22.49C24  C22.50C24  Q8⋊3Q8  C22.53C24

Matrix representation of C2×C4○D4 ►in GL3(𝔽5) generated by

400
040
004
,
400
030
003
,
400
001
040
,
100
001
010
G:=sub<GL(3,GF(5))| [4,0,0,0,4,0,0,0,4],[4,0,0,0,3,0,0,0,3],[4,0,0,0,0,4,0,1,0],[1,0,0,0,0,1,0,1,0] >;
 

C2×C4○D4 in GAP, Magma, Sage, TeX 

C_2\times C_4\circ D_4
 
% in TeX
 
G:=Group("C2xC4oD4");
 
// GroupNames label
 
G:=SmallGroup(32,48);
 
// by ID
 
G=gap.SmallGroup(32,48);
 
# by ID
 
G:=PCGroup([5,-2,2,2,2,-2,181,72]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^2=b^4=d^2=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^2*c>;
 
// generators/relations
 

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Character table of C2×C4○D4 in TeX

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