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G = C40⋊C4order 160 = 25·5

2nd semidirect product of C40 and C4 acting faithfully

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C82F5, C402C4, D10.8D4, Dic5.2Q8, D5.1SD16, C5⋊(C4.Q8), C52C85C4, C4⋊F5.3C2, C4.8(C2×F5), C20.8(C2×C4), (C8×D5).5C2, C2.4(C4⋊F5), C10.1(C4⋊C4), (C4×D5).25C22, SmallGroup(160,68)

Series: Derived Chief Lower central Upper central

C1C20 — C40⋊C4
C1C5C10D10C4×D5C4⋊F5 — C40⋊C4
C5C10C20 — C40⋊C4
C1C2C4C8

Generators and relations for C40⋊C4
 G = < a,b | a40=b4=1, bab-1=a3 >

5C2
5C2
5C4
5C22
20C4
20C4
5C2×C4
5C8
10C2×C4
10C2×C4
4F5
4F5
5C2×C8
5C4⋊C4
5C4⋊C4
2C2×F5
2C2×F5
5C4.Q8

Character table of C40⋊C4

 class 12A2B2C4A4B4C4D4E4F58A8B8C8D1020A20B40A40B40C40D
 size 11552102020202042210104444444
ρ11111111111111111111111    trivial
ρ2111111-11-111-1-1-1-1111-1-1-1-1    linear of order 2
ρ3111111-1-1-1-1111111111111    linear of order 2
ρ41111111-11-11-1-1-1-1111-1-1-1-1    linear of order 2
ρ511-1-11-1ii-i-i111-1-11111111    linear of order 4
ρ611-1-11-1-iii-i1-1-111111-1-1-1-1    linear of order 4
ρ711-1-11-1-i-iii111-1-11111111    linear of order 4
ρ811-1-11-1i-i-ii1-1-111111-1-1-1-1    linear of order 4
ρ92222-2-20000200002-2-20000    orthogonal lifted from D4
ρ1022-2-2-220000200002-2-20000    symplectic lifted from Q8, Schur index 2
ρ112-22-20000002-2--2--2-2-200--2--2-2-2    complex lifted from SD16
ρ122-22-20000002--2-2-2--2-200-2-2--2--2    complex lifted from SD16
ρ132-2-220000002-2--2-2--2-200--2--2-2-2    complex lifted from SD16
ρ142-2-220000002--2-2--2-2-200-2-2--2--2    complex lifted from SD16
ρ154400400000-1-4-400-1-1-11111    orthogonal lifted from C2×F5
ρ164400400000-14400-1-1-1-1-1-1-1    orthogonal lifted from F5
ρ174400-400000-10000-111--5-5-5--5    complex lifted from C4⋊F5
ρ184400-400000-10000-111-5--5--5-5    complex lifted from C4⋊F5
ρ194-400000000-1-2-22-2001--5-5ζ87ζ5487ζ58785ζ5485ζ587ζ5487ζ585ζ5485ζ585ζ83ζ5383ζ52838ζ538ζ5283ζ5383ζ528ζ538ζ528    complex faithful
ρ204-400000000-12-2-2-2001-5--5ζ83ζ5383ζ52838ζ538ζ5283ζ5383ζ528ζ538ζ528ζ87ζ5487ζ58785ζ5485ζ587ζ5487ζ585ζ5485ζ585    complex faithful
ρ214-400000000-12-2-2-2001--5-583ζ5383ζ528ζ538ζ528ζ83ζ5383ζ52838ζ538ζ5287ζ5487ζ585ζ5485ζ585ζ87ζ5487ζ58785ζ5485ζ5    complex faithful
ρ224-400000000-1-2-22-2001-5--587ζ5487ζ585ζ5485ζ585ζ87ζ5487ζ58785ζ5485ζ583ζ5383ζ528ζ538ζ528ζ83ζ5383ζ52838ζ538ζ52    complex faithful

Smallest permutation representation of C40⋊C4
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)
(2 28 10 4)(3 15 19 7)(5 29 37 13)(6 16)(8 30 24 22)(9 17 33 25)(11 31)(12 18 20 34)(14 32 38 40)(23 35 39 27)(26 36)

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

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

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

C40⋊C4 is a maximal subgroup of
C804C4  C805C4  D10.D8  D401C4  (C2×C8)⋊6F5  M4(2)⋊1F5  D40⋊C4  SD16×F5  Dic20⋊C4  Dic5.Dic6  C120⋊C4
C40⋊C4 is a maximal quotient of
C804C4  C805C4  C402C8  C20.26M4(2)  D10.10D8  Dic5.Dic6  C120⋊C4

Matrix representation of C40⋊C4 in GL4(𝔽3) generated by

0022
1000
2102
2200
,
2010
0010
0200
0001
G:=sub<GL(4,GF(3))| [0,1,2,2,0,0,1,2,2,0,0,0,2,0,2,0],[2,0,0,0,0,0,2,0,1,1,0,0,0,0,0,1] >;

C40⋊C4 in GAP, Magma, Sage, TeX

C_{40}\rtimes C_4
% in TeX

G:=Group("C40:C4");
// GroupNames label

G:=SmallGroup(160,68);
// by ID

G=gap.SmallGroup(160,68);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,24,121,55,579,69,2309,1169]);
// Polycyclic

G:=Group<a,b|a^40=b^4=1,b*a*b^-1=a^3>;
// generators/relations

Export

Subgroup lattice of C40⋊C4 in TeX
Character table of C40⋊C4 in TeX

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