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G = C4⋊C8order 32 = 25

The semidirect product of C4 and C8 acting via C8/C4=C2

p-group, metacyclic, nilpotent (class 2), monomial

Aliases: C4⋊C8, C4.4Q8, C4.18D4, C42.2C2, C2.3M4(2), (C2×C4).4C4, C2.2(C2×C8), (C2×C8).2C2, C2.2(C4⋊C4), C22.9(C2×C4), (C2×C4).33C22, SmallGroup(32,12)

Series: Derived Chief Lower central Upper central Jennings

C1C2 — C4⋊C8
C1C2C4C2×C4C42 — C4⋊C8
C1C2 — C4⋊C8
C1C2×C4 — C4⋊C8
C1C2C2C2×C4 — C4⋊C8

Generators and relations for C4⋊C8
 G = < a,b | a4=b8=1, bab-1=a-1 >

2C4
2C8
2C8

Character table of C4⋊C8

 class 12A2B2C4A4B4C4D4E4F4G4H8A8B8C8D8E8F8G8H
 size 11111111222222222222
ρ111111111111111111111    trivial
ρ211111111-1-1-1-1-1-1-1-11111    linear of order 2
ρ3111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ411111111-1-1-1-11111-1-1-1-1    linear of order 2
ρ51111-1-1-1-11-11-1-ii-ii-iii-i    linear of order 4
ρ61111-1-1-1-1-11-11i-ii-i-iii-i    linear of order 4
ρ71111-1-1-1-11-11-1i-ii-ii-i-ii    linear of order 4
ρ81111-1-1-1-1-11-11-ii-iii-i-ii    linear of order 4
ρ91-11-1-i-iii-i-1i1ζ85ζ87ζ8ζ83ζ87ζ85ζ8ζ83    linear of order 8
ρ101-11-1-i-iiii1-i-1ζ8ζ83ζ85ζ87ζ87ζ85ζ8ζ83    linear of order 8
ρ111-11-1-i-iii-i-1i1ζ8ζ83ζ85ζ87ζ83ζ8ζ85ζ87    linear of order 8
ρ121-11-1-i-iiii1-i-1ζ85ζ87ζ8ζ83ζ83ζ8ζ85ζ87    linear of order 8
ρ131-11-1ii-i-i-i1i-1ζ83ζ8ζ87ζ85ζ85ζ87ζ83ζ8    linear of order 8
ρ141-11-1ii-i-i-i1i-1ζ87ζ85ζ83ζ8ζ8ζ83ζ87ζ85    linear of order 8
ρ151-11-1ii-i-ii-1-i1ζ87ζ85ζ83ζ8ζ85ζ87ζ83ζ8    linear of order 8
ρ161-11-1ii-i-ii-1-i1ζ83ζ8ζ87ζ85ζ8ζ83ζ87ζ85    linear of order 8
ρ1722-2-22-22-2000000000000    orthogonal lifted from D4
ρ1822-2-2-22-22000000000000    symplectic lifted from Q8, Schur index 2
ρ192-2-222i-2i-2i2i000000000000    complex lifted from M4(2)
ρ202-2-22-2i2i2i-2i000000000000    complex lifted from M4(2)

Smallest permutation representation of C4⋊C8
Regular action on 32 points
Generators in S32
(1 23 31 11)(2 12 32 24)(3 17 25 13)(4 14 26 18)(5 19 27 15)(6 16 28 20)(7 21 29 9)(8 10 30 22)
(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)

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

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

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

C4⋊C8 is a maximal subgroup of
C4.D8  C4.10D8  C4.6Q16  C4⋊M4(2)  C42.6C22  C42.7C22  C8×D4  C89D4  C86D4  C8×Q8  C4⋊D8  C4⋊SD16  D4.D4  C42Q16  D4.2D4  Q8.D4  D4⋊Q8  Q8⋊Q8  D42Q8  C4.Q16  D4.Q8  Q8.Q8  C32⋊C4⋊C8  C4.4PSU3(𝔽2)  (C3×C12)⋊4C8  C325(C4⋊C8)  C4⋊F9
 C4p⋊C8: C82C8  C81C8  C12⋊C8  C203C8  C20⋊C8  C28⋊C8  C44⋊C8  C523C8 ...
 C2p.M4(2): D4⋊C8  Q8⋊C8  C42.12C4  C42.6C4  C84Q8  Dic3⋊C8  C20.8Q8  Dic5⋊C8 ...
C4⋊C8 is a maximal quotient of
C22.7C42  Dic5⋊C8  C32⋊C4⋊C8  C4.4PSU3(𝔽2)  (C3×C12)⋊4C8  C325(C4⋊C8)  C4⋊F9  Dic13⋊C8
 C4p⋊C8: C82C8  C81C8  C12⋊C8  C203C8  C20⋊C8  C28⋊C8  C44⋊C8  C523C8 ...
 C4p.Q8: C4⋊C16  C8.C8  Dic3⋊C8  C20.8Q8  Dic7⋊C8  Dic11⋊C8  C52.8Q8 ...

Matrix representation of C4⋊C8 in GL3(𝔽17) generated by

100
01615
011
,
200
0160
011
G:=sub<GL(3,GF(17))| [1,0,0,0,16,1,0,15,1],[2,0,0,0,16,1,0,0,1] >;

C4⋊C8 in GAP, Magma, Sage, TeX

C_4\rtimes C_8
% in TeX

G:=Group("C4:C8");
// GroupNames label

G:=SmallGroup(32,12);
// by ID

G=gap.SmallGroup(32,12);
# by ID

G:=PCGroup([5,-2,2,-2,2,-2,40,61,26,58]);
// Polycyclic

G:=Group<a,b|a^4=b^8=1,b*a*b^-1=a^-1>;
// generators/relations

Export

Subgroup lattice of C4⋊C8 in TeX
Character table of C4⋊C8 in TeX

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