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G = C8⋊C4order 32 = 25

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

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

Aliases: C83C4, 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

C1C2 — C8⋊C4
C1C2C22C2×C4C42 — C8⋊C4
C1C2 — C8⋊C4
C1C2×C4 — C8⋊C4
C1C2C2C2×C4 — C8⋊C4

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

2C4
2C4

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

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

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

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

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