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G = C8⋊C4  order 32 = 25

3rd semidirect product of C8 and C4 acting via C4/C2=C2

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

Aliases: C8⋊3C4, C42.1C2, C2.2C42, C2.1M4(2), (C2×C8).7C2, (C2×C4).2C4, C4.11(C2×C4), C22.7(C2×C4), (C2×C4).31C22, SmallGroup(32,4)

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

Derived series
C1 — C2 — C8⋊C4
Chief series
C1 — C2 — C22 — C2×C4 — C42 — C8⋊C4
Lower central
C1 — C2 — C8⋊C4
Upper central
C1 — C2×C4 — C8⋊C4
Jennings
C1 — C2 — C2 — C2×C4 — C8⋊C4

Generators and relations for C8⋊C4
 G = < a,b | a8=b4=1, bab-1=a5 >

C1
C2
C2
C2
C4
C22
C4
2C4
2C4
C8
C8
C2×C4
C8
C2×C4
C2×C4
C8
C2×C8
C2×C8
C42
C8⋊C4

Character table of C8⋊C4

 class 12A2B2C4A4B4C4D4E4F4G4H8A8B8C8D8E8F8G8H
 size 11111111222222222222
ρ111111111111111111111    trivial
ρ211111111-1-1-1-1-1111-1-11-1    linear of order 2
ρ3111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ411111111-1-1-1-11-1-1-111-11    linear of order 2
ρ51-11-1-1-111-ii-ii-1i-ii-11-i1    linear of order 4
ρ61-11-111-1-1i-i-iii-1-11-ii1-i    linear of order 4
ρ71-11-1-1-111-ii-ii1-ii-i1-1i-1    linear of order 4
ρ81-11-111-1-1i-i-ii-i11-1i-i-1i    linear of order 4
ρ91111-1-1-1-111-1-1ii-i-i-i-iii    linear of order 4
ρ101-11-1-1-111i-ii-i1i-ii1-1-i-1    linear of order 4
ρ111-11-111-1-1-iii-i-i-1-11i-i1i    linear of order 4
ρ121111-1-1-1-1-1-111-ii-i-iiii-i    linear of order 4
ρ131111-1-1-1-111-1-1-i-iiiii-i-i    linear of order 4
ρ141-11-1-1-111i-ii-i-1-ii-i-11i1    linear of order 4
ρ151-11-111-1-1-iii-ii11-1-ii-1-i    linear of order 4
ρ161111-1-1-1-1-1-111i-iii-i-i-ii    linear of order 4
ρ172-2-222i-2i-2i2i000000000000    complex lifted from M4(2)
ρ1822-2-2-2i2i-2i2i000000000000    complex lifted from M4(2)
ρ192-2-22-2i2i2i-2i000000000000    complex lifted from M4(2)
ρ2022-2-22i-2i2i-2i000000000000    complex lifted from M4(2)

Smallest permutation representation of C8⋊C4
►Regular action on 32 points
Generators in S32
(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 10 32 19)(2 15 25 24)(3 12 26 21)(4 9 27 18)(5 14 28 23)(6 11 29 20)(7 16 30 17)(8 13 31 22)
 
G:=sub<Sym(32)| (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), (1,10,32,19)(2,15,25,24)(3,12,26,21)(4,9,27,18)(5,14,28,23)(6,11,29,20)(7,16,30,17)(8,13,31,22)>;
 
G:=Group( (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), (1,10,32,19)(2,15,25,24)(3,12,26,21)(4,9,27,18)(5,14,28,23)(6,11,29,20)(7,16,30,17)(8,13,31,22) );
 
G=PermutationGroup([[(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)], [(1,10,32,19),(2,15,25,24),(3,12,26,21),(4,9,27,18),(5,14,28,23),(6,11,29,20),(7,16,30,17),(8,13,31,22)]])
 

C8⋊C4 is a maximal subgroup of
 C42.C22  C42.2C22  C42.6C4  C42.7C22  C8⋊9D4  SD16⋊C4  Q16⋊C4  D8⋊C4  C8.26D4  C8⋊4Q8  C42.28C22  C42.29C22  C42.30C22  C8⋊3D4  C8.2D4  C8⋊Q8  (C3×C24)⋊C4  C32⋊2C8⋊C4
 C8p⋊C4: C16⋊C4  C24⋊C4  C40⋊8C4  C8⋊F5  C56⋊C4  C88⋊C4  C104⋊8C4  C104⋊C4 ...
 C2p.C42: C4×M4(2)  C8○2M4(2)  C42.S3  C42.D5  C10.C42  C42.D7  C42.D11  C26.7C42 ...
C8⋊C4 is a maximal quotient of
 C22.7C42  (C3×C24)⋊C4  C32⋊2C8⋊C4
 C8p⋊C4: C16⋊C4  C24⋊C4  C40⋊8C4  C8⋊F5  C56⋊C4  C88⋊C4  C104⋊8C4  C104⋊C4 ...
 C2p.M4(2): C8⋊C8  C42.S3  C42.D5  C10.C42  C42.D7  C42.D11  C26.7C42  C26.C42 ...

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

1300
0510
0912
,
1300
01315
0164
G:=sub<GL(3,GF(17))| [13,0,0,0,5,9,0,10,12],[13,0,0,0,13,16,0,15,4] >;
 

C8⋊C4 in GAP, Magma, Sage, TeX 

C_8\rtimes C_4
 
% in TeX
 
G:=Group("C8:C4");
 
// GroupNames label
 
G:=SmallGroup(32,4);
 
// by ID
 
G=gap.SmallGroup(32,4);
 
# by ID
 
G:=PCGroup([5,-2,2,-2,2,-2,20,181,46,72]);
 
// Polycyclic
 
G:=Group<a,b|a^8=b^4=1,b*a*b^-1=a^5>;
 
// generators/relations
 

Export

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

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