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G = C4○D8order 32 = 25

Central product of C4 and D8

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

Aliases: C4D8, C4Q16, D83C2, C4SD16, Q163C2, C4.20D4, SD163C2, C4.4C23, C8.6C22, C22.1D4, D4.2C22, Q8.2C22, (C2×C8)⋊4C2, C4○D41C2, C2.14(C2×D4), (C2×C4).29C22, SmallGroup(32,42)

Series: Derived Chief Lower central Upper central Jennings

C1C4 — C4○D8
C1C2C4C2×C4C4○D4 — C4○D8
C1C2C4 — C4○D8
C1C4C2×C4 — C4○D8
C1C2C2C4 — C4○D8

Generators and relations for C4○D8
 G = < a,b,c | a4=c2=1, b4=a2, ab=ba, ac=ca, cbc=a2b3 >

2C2
4C2
4C2
2C22
2C22
2C4
2C4
2C2×C4
2D4
2C2×C4
2D4

Character table of C4○D8

 class 12A2B2C2D4A4B4C4D4E8A8B8C8D
 size 11244112442222
ρ111111111111111    trivial
ρ21111-1111-11-1-1-1-1    linear of order 2
ρ311-111-1-11-1-1-1-111    linear of order 2
ρ411-11-1-1-111-111-1-1    linear of order 2
ρ511-1-11-1-11-1111-1-1    linear of order 2
ρ611-1-1-1-1-1111-1-111    linear of order 2
ρ7111-111111-1-1-1-1-1    linear of order 2
ρ8111-1-1111-1-11111    linear of order 2
ρ922-20022-2000000    orthogonal lifted from D4
ρ1022200-2-2-2000000    orthogonal lifted from D4
ρ112-2000-2i2i000--2-2-22    complex faithful
ρ122-2000-2i2i000-2--22-2    complex faithful
ρ132-20002i-2i000-2--2-22    complex faithful
ρ142-20002i-2i000--2-22-2    complex faithful

Permutation representations of C4○D8
On 16 points - transitive group 16T44
Generators in S16
(1 10 5 14)(2 11 6 15)(3 12 7 16)(4 13 8 9)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(2 8)(3 7)(4 6)(9 11)(12 16)(13 15)

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

G:=Group( (1,10,5,14)(2,11,6,15)(3,12,7,16)(4,13,8,9), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (2,8)(3,7)(4,6)(9,11)(12,16)(13,15) );

G=PermutationGroup([(1,10,5,14),(2,11,6,15),(3,12,7,16),(4,13,8,9)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(2,8),(3,7),(4,6),(9,11),(12,16),(13,15)])

G:=TransitiveGroup(16,44);

On 16 points - transitive group 16T47
Generators in S16
(1 3 5 7)(2 4 6 8)(9 15 13 11)(10 16 14 12)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(1 9)(2 16)(3 15)(4 14)(5 13)(6 12)(7 11)(8 10)

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

G:=Group( (1,3,5,7)(2,4,6,8)(9,15,13,11)(10,16,14,12), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,9)(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10) );

G=PermutationGroup([(1,3,5,7),(2,4,6,8),(9,15,13,11),(10,16,14,12)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(1,9),(2,16),(3,15),(4,14),(5,13),(6,12),(7,11),(8,10)])

G:=TransitiveGroup(16,47);

C4○D8 is a maximal subgroup of
D8.C4  D82C4  C8○D8  C8.26D4  Q32⋊C2  D8⋊C22  C4.6S4  C32⋊D85C2  C62.13D4
 D8p⋊C2: C4○D16  C16⋊C22  C4○D24  D24⋊C2  D407C2  Q8.D10  D567C2  Q8.D14 ...
 D4.D2p: D4○D8  D4○SD16  Q8○D8  D83S3  Q8.7D6  Q8.13D6  D83D5  SD163D5 ...
C4○D8 is a maximal quotient of
C23.24D4  C23.25D4  C4×D8  C4×SD16  C4×Q16  D4⋊D4  C88D4  C87D4  D4.Q8  Q8.Q8  C23.19D4  C23.20D4  C42.78C22  C8.5Q8  C32⋊D85C2  C62.13D4
 D4.D2p: D4.7D4  D4.2D4  D83S3  Q8.7D6  Q8.13D6  D83D5  SD163D5  D4.8D10 ...
 Q8.D2p: Q8.D4  D24⋊C2  Q8.D10  Q8.D14  D885C2  D104⋊C2 ...
 C4p.D4: C8.18D4  C8.12D4  C4○D24  D407C2  D567C2  D887C2  D1047C2 ...

Matrix representation of C4○D8 in GL2(𝔽17) generated by

40
04
,
06
146
,
06
30
G:=sub<GL(2,GF(17))| [4,0,0,4],[0,14,6,6],[0,3,6,0] >;

C4○D8 in GAP, Magma, Sage, TeX

C_4\circ D_8
% in TeX

G:=Group("C4oD8");
// GroupNames label

G:=SmallGroup(32,42);
// by ID

G=gap.SmallGroup(32,42);
# by ID

G:=PCGroup([5,-2,2,2,-2,-2,101,72,483,248,58]);
// Polycyclic

G:=Group<a,b,c|a^4=c^2=1,b^4=a^2,a*b=b*a,a*c=c*a,c*b*c=a^2*b^3>;
// generators/relations

Export

Subgroup lattice of C4○D8 in TeX
Character table of C4○D8 in TeX

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