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G = C4×Q8order 32 = 25

Direct product of C4 and Q8

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

Aliases: C4×Q8, C42.3C2, C22.8C23, C43(C4⋊C4), C4⋊C4.6C2, C4.4(C2×C4), C2.2(C2×Q8), (C2×Q8).5C2, C2.3(C4○D4), C2.5(C22×C4), (C2×C4).12C22, SmallGroup(32,26)

Series: Derived Chief Lower central Upper central Jennings

C1C2 — C4×Q8
C1C2C22C2×C4C42 — C4×Q8
C1C2 — C4×Q8
C1C2×C4 — C4×Q8
C1C22 — C4×Q8

Generators and relations for C4×Q8
 G = < a,b,c | a4=b4=1, c2=b2, ab=ba, ac=ca, cbc-1=b-1 >

2C4
2C4
2C4

Character table of C4×Q8

 class 12A2B2C4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P
 size 11111111222222222222
ρ111111111111111111111    trivial
ρ211111111-1-1-111-1-1-1-1-111    linear of order 2
ρ311111111-111-1-1-111-1-1-1-1    linear of order 2
ρ4111111111-1-1-1-11-1-111-1-1    linear of order 2
ρ51111-1-1-1-1-1-111-11-11-111-1    linear of order 2
ρ61111-1-1-1-111-11-1-11-11-11-1    linear of order 2
ρ71111-1-1-1-11-11-11-1-111-1-11    linear of order 2
ρ81111-1-1-1-1-11-1-1111-1-11-11    linear of order 2
ρ91-11-1-ii-iii1ii-1-1-1-i-i1-i1    linear of order 4
ρ101-11-1-ii-ii-i-1-ii-111ii-1-i1    linear of order 4
ρ111-11-1i-ii-ii-1i-i-111-i-i-1i1    linear of order 4
ρ121-11-1i-ii-i-i1-i-i-1-1-1ii1i1    linear of order 4
ρ131-11-1i-ii-i-i-1ii1-11-ii1-i-1    linear of order 4
ρ141-11-1i-ii-ii1-ii11-1i-i-1-i-1    linear of order 4
ρ151-11-1-ii-ii-i1i-i11-1-ii-1i-1    linear of order 4
ρ161-11-1-ii-iii-1-i-i1-11i-i1i-1    linear of order 4
ρ1722-2-2-2-222000000000000    symplectic lifted from Q8, Schur index 2
ρ1822-2-222-2-2000000000000    symplectic lifted from Q8, Schur index 2
ρ192-2-222i-2i-2i2i000000000000    complex lifted from C4○D4
ρ202-2-22-2i2i2i-2i000000000000    complex lifted from C4○D4

Smallest permutation representation of C4×Q8
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 17 9 14)(2 18 10 15)(3 19 11 16)(4 20 12 13)(5 27 30 24)(6 28 31 21)(7 25 32 22)(8 26 29 23)
(1 28 9 21)(2 25 10 22)(3 26 11 23)(4 27 12 24)(5 13 30 20)(6 14 31 17)(7 15 32 18)(8 16 29 19)

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

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

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

Matrix representation of C4×Q8 in GL3(𝔽5) generated by

200
030
003
,
100
030
002
,
400
003
030
G:=sub<GL(3,GF(5))| [2,0,0,0,3,0,0,0,3],[1,0,0,0,3,0,0,0,2],[4,0,0,0,0,3,0,3,0] >;

C4×Q8 in GAP, Magma, Sage, TeX

C_4\times Q_8
% in TeX

G:=Group("C4xQ8");
// GroupNames label

G:=SmallGroup(32,26);
// by ID

G=gap.SmallGroup(32,26);
# by ID

G:=PCGroup([5,-2,2,2,-2,2,80,101,46,102]);
// Polycyclic

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

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