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G = C5×Q8  order 40 = 23·5

Direct product of C5 and Q8

direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary

Aliases: C5×Q8, C4.C10, C20.3C2, C10.7C22, C2.2(C2×C10), SmallGroup(40,11)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2 — C5×Q8
Chief series
C1 — C2 — C10 — C20 — C5×Q8
Lower central
C1 — C2 — C5×Q8
Upper central
C1 — C10 — C5×Q8

Generators and relations for C5×Q8
 G = < a,b,c | a5=b4=1, c2=b2, ab=ba, ac=ca, cbc-1=b-1 >

C1
C2
C5
C4
C4
C4
C10
Q8
C20
C20
C20
C5×Q8

Character table of C5×Q8

 class 124A4B4C5A5B5C5D10A10B10C10D20A20B20C20D20E20F20G20H20I20J20K20L
 size 1122211111111222222222222
ρ11111111111111111111111111    trivial
ρ2111-1-1111111111-1-1-1-1-1-1-1111-1    linear of order 2
ρ311-11-111111111-1-1-11111-1-1-1-1-1    linear of order 2
ρ411-1-1111111111-111-1-1-1-11-1-1-11    linear of order 2
ρ5111-1-1ζ52ζ53ζ54ζ5ζ54ζ53ζ5ζ52ζ54-ζ54-ζ5-ζ5-ζ52-ζ53-ζ54-ζ52ζ5ζ52ζ53-ζ53    linear of order 10
ρ611-1-11ζ52ζ53ζ54ζ5ζ54ζ53ζ5ζ52-ζ54ζ54ζ5-ζ5-ζ52-ζ53-ζ54ζ52-ζ5-ζ52-ζ53ζ53    linear of order 10
ρ711111ζ5ζ54ζ52ζ53ζ52ζ54ζ53ζ5ζ52ζ52ζ53ζ53ζ5ζ54ζ52ζ5ζ53ζ5ζ54ζ54    linear of order 5
ρ811-11-1ζ54ζ5ζ53ζ52ζ53ζ5ζ52ζ54-ζ53-ζ53-ζ52ζ52ζ54ζ5ζ53-ζ54-ζ52-ζ54-ζ5-ζ5    linear of order 10
ρ911-1-11ζ5ζ54ζ52ζ53ζ52ζ54ζ53ζ5-ζ52ζ52ζ53-ζ53-ζ5-ζ54-ζ52ζ5-ζ53-ζ5-ζ54ζ54    linear of order 10
ρ10111-1-1ζ54ζ5ζ53ζ52ζ53ζ5ζ52ζ54ζ53-ζ53-ζ52-ζ52-ζ54-ζ5-ζ53-ζ54ζ52ζ54ζ5-ζ5    linear of order 10
ρ1111111ζ53ζ52ζ5ζ54ζ5ζ52ζ54ζ53ζ5ζ5ζ54ζ54ζ53ζ52ζ5ζ53ζ54ζ53ζ52ζ52    linear of order 5
ρ1211-11-1ζ53ζ52ζ5ζ54ζ5ζ52ζ54ζ53-ζ5-ζ5-ζ54ζ54ζ53ζ52ζ5-ζ53-ζ54-ζ53-ζ52-ζ52    linear of order 10
ρ1311-11-1ζ5ζ54ζ52ζ53ζ52ζ54ζ53ζ5-ζ52-ζ52-ζ53ζ53ζ5ζ54ζ52-ζ5-ζ53-ζ5-ζ54-ζ54    linear of order 10
ρ14111-1-1ζ5ζ54ζ52ζ53ζ52ζ54ζ53ζ5ζ52-ζ52-ζ53-ζ53-ζ5-ζ54-ζ52-ζ5ζ53ζ5ζ54-ζ54    linear of order 10
ρ15111-1-1ζ53ζ52ζ5ζ54ζ5ζ52ζ54ζ53ζ5-ζ5-ζ54-ζ54-ζ53-ζ52-ζ5-ζ53ζ54ζ53ζ52-ζ52    linear of order 10
ρ1611-1-11ζ53ζ52ζ5ζ54ζ5ζ52ζ54ζ53-ζ5ζ5ζ54-ζ54-ζ53-ζ52-ζ5ζ53-ζ54-ζ53-ζ52ζ52    linear of order 10
ρ1711111ζ54ζ5ζ53ζ52ζ53ζ5ζ52ζ54ζ53ζ53ζ52ζ52ζ54ζ5ζ53ζ54ζ52ζ54ζ5ζ5    linear of order 5
ρ1811111ζ52ζ53ζ54ζ5ζ54ζ53ζ5ζ52ζ54ζ54ζ5ζ5ζ52ζ53ζ54ζ52ζ5ζ52ζ53ζ53    linear of order 5
ρ1911-11-1ζ52ζ53ζ54ζ5ζ54ζ53ζ5ζ52-ζ54-ζ54-ζ5ζ5ζ52ζ53ζ54-ζ52-ζ5-ζ52-ζ53-ζ53    linear of order 10
ρ2011-1-11ζ54ζ5ζ53ζ52ζ53ζ5ζ52ζ54-ζ53ζ53ζ52-ζ52-ζ54-ζ5-ζ53ζ54-ζ52-ζ54-ζ5ζ5    linear of order 10
ρ212-20002222-2-2-2-2000000000000    symplectic lifted from Q8, Schur index 2
ρ222-20002ζ542ζ52ζ532ζ52-2ζ53-2ζ5-2ζ52-2ζ54000000000000    complex faithful
ρ232-20002ζ52ζ542ζ522ζ53-2ζ52-2ζ54-2ζ53-2ζ5000000000000    complex faithful
ρ242-20002ζ522ζ532ζ542ζ5-2ζ54-2ζ53-2ζ5-2ζ52000000000000    complex faithful
ρ252-20002ζ532ζ522ζ52ζ54-2ζ5-2ζ52-2ζ54-2ζ53000000000000    complex faithful

Smallest permutation representation of C5×Q8
►Regular action on 40 points
Generators in S40
(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 33 34 35)(36 37 38 39 40)

(1 23 13 17)(2 24 14 18)(3 25 15 19)(4 21 11 20)(5 22 12 16)(6 30 40 31)(7 26 36 32)(8 27 37 33)(9 28 38 34)(10 29 39 35)

(1 33 13 27)(2 34 14 28)(3 35 15 29)(4 31 11 30)(5 32 12 26)(6 21 40 20)(7 22 36 16)(8 23 37 17)(9 24 38 18)(10 25 39 19)
 
G:=sub<Sym(40)| (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)(31,32,33,34,35)(36,37,38,39,40), (1,23,13,17)(2,24,14,18)(3,25,15,19)(4,21,11,20)(5,22,12,16)(6,30,40,31)(7,26,36,32)(8,27,37,33)(9,28,38,34)(10,29,39,35), (1,33,13,27)(2,34,14,28)(3,35,15,29)(4,31,11,30)(5,32,12,26)(6,21,40,20)(7,22,36,16)(8,23,37,17)(9,24,38,18)(10,25,39,19)>;
 
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)(31,32,33,34,35)(36,37,38,39,40), (1,23,13,17)(2,24,14,18)(3,25,15,19)(4,21,11,20)(5,22,12,16)(6,30,40,31)(7,26,36,32)(8,27,37,33)(9,28,38,34)(10,29,39,35), (1,33,13,27)(2,34,14,28)(3,35,15,29)(4,31,11,30)(5,32,12,26)(6,21,40,20)(7,22,36,16)(8,23,37,17)(9,24,38,18)(10,25,39,19) );
 
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),(31,32,33,34,35),(36,37,38,39,40)], [(1,23,13,17),(2,24,14,18),(3,25,15,19),(4,21,11,20),(5,22,12,16),(6,30,40,31),(7,26,36,32),(8,27,37,33),(9,28,38,34),(10,29,39,35)], [(1,33,13,27),(2,34,14,28),(3,35,15,29),(4,31,11,30),(5,32,12,26),(6,21,40,20),(7,22,36,16),(8,23,37,17),(9,24,38,18),(10,25,39,19)]])
 

C5×Q8 is a maximal subgroup of   Q8⋊D5  C5⋊Q16  Q8⋊2D5  C4.F11
C5×Q8 is a maximal quotient of   C4.F11

Matrix representation of C5×Q8 ►in GL2(𝔽11) generated by

50
05
,
47
77
,
010
10
G:=sub<GL(2,GF(11))| [5,0,0,5],[4,7,7,7],[0,1,10,0] >;
 

C5×Q8 in GAP, Magma, Sage, TeX 

C_5\times Q_8
 
% in TeX
 
G:=Group("C5xQ8");
 
// GroupNames label
 
G:=SmallGroup(40,11);
 
// by ID
 
G=gap.SmallGroup(40,11);
 
# by ID
 
G:=PCGroup([4,-2,-2,-5,-2,80,177,85]);
 
// Polycyclic
 
G:=Group<a,b,c|a^5=b^4=1,c^2=b^2,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
 
// generators/relations
 

Export

Subgroup lattice of C5×Q8 in TeX
Character table of C5×Q8 in TeX

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