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G = C40.C4order 160 = 25·5

2nd non-split extension by C40 of C4 acting faithfully

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C8.2F5, C40.2C4, D10.1Q8, Dic5.10D4, C52C8.3C4, (C8×D5).6C2, C2.6(C4⋊F5), C10.3(C4⋊C4), C4.10(C2×F5), C51(C8.C4), C4.F5.1C2, C20.10(C2×C4), (C4×D5).27C22, SmallGroup(160,70)

Series: Derived Chief Lower central Upper central

C1C20 — C40.C4
C1C5C10Dic5C4×D5C4.F5 — C40.C4
C5C10C20 — C40.C4
C1C2C4C8

Generators and relations for C40.C4
 G = < a,b | a40=1, b4=a20, bab-1=a3 >

10C2
5C22
5C4
2D5
5C8
5C2×C4
10C8
10C8
5C2×C8
5M4(2)
5M4(2)
2C5⋊C8
2C5⋊C8
5C8.C4

Character table of C40.C4

 class 12A2B4A4B4C58A8B8C8D8E8F8G8H1020A20B40A40B40C40D
 size 11102554221010202020204444444
ρ11111111111111111111111    trivial
ρ21111111-1-1-1-1-11-11111-1-1-1-1    linear of order 2
ρ31111111-1-1-1-11-11-1111-1-1-1-1    linear of order 2
ρ411111111111-1-1-1-11111111    linear of order 2
ρ511-11-1-11-1-111i-i-ii111-1-1-1-1    linear of order 4
ρ611-11-1-11-1-111-iii-i111-1-1-1-1    linear of order 4
ρ711-11-1-1111-1-1-i-iii1111111    linear of order 4
ρ811-11-1-1111-1-1ii-i-i1111111    linear of order 4
ρ922-2-2222000000002-2-20000    orthogonal lifted from D4
ρ10222-2-2-22000000002-2-20000    symplectic lifted from Q8, Schur index 2
ρ112-200-2i2i2-2--2-220000-200--2-2-2--2    complex lifted from C8.C4
ρ122-2002i-2i2-2--22-20000-200--2-2-2--2    complex lifted from C8.C4
ρ132-2002i-2i2--2-2-220000-200-2--2--2-2    complex lifted from C8.C4
ρ142-200-2i2i2--2-22-20000-200-2--2--2-2    complex lifted from C8.C4
ρ15440400-1-4-4000000-1-1-11111    orthogonal lifted from C2×F5
ρ16440400-144000000-1-1-1-1-1-1-1    orthogonal lifted from F5
ρ17440-400-100000000-111--5--5-5-5    complex lifted from C4⋊F5
ρ18440-400-100000000-111-5-5--5--5    complex lifted from C4⋊F5
ρ194-40000-1-2-22-20000001--5-587ζ5487ζ585ζ5485ζ585ζ83ζ5383ζ52838ζ538ζ5283ζ5383ζ528ζ538ζ528ζ87ζ5487ζ58785ζ5485ζ5    complex faithful
ρ204-40000-12-2-2-20000001-5--583ζ5383ζ528ζ538ζ528ζ87ζ5487ζ58785ζ5485ζ587ζ5487ζ585ζ5485ζ585ζ83ζ5383ζ52838ζ538ζ52    complex faithful
ρ214-40000-1-2-22-20000001-5--5ζ87ζ5487ζ58785ζ5485ζ583ζ5383ζ528ζ538ζ528ζ83ζ5383ζ52838ζ538ζ5287ζ5487ζ585ζ5485ζ585    complex faithful
ρ224-40000-12-2-2-20000001--5-5ζ83ζ5383ζ52838ζ538ζ5287ζ5487ζ585ζ5485ζ585ζ87ζ5487ζ58785ζ5485ζ583ζ5383ζ528ζ538ζ528    complex faithful

Smallest permutation representation of C40.C4
On 80 points
Generators in S80
(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)(41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80)
(1 76 31 46 21 56 11 66)(2 63 40 49 22 43 20 69)(3 50 9 52 23 70 29 72)(4 77 18 55 24 57 38 75)(5 64 27 58 25 44 7 78)(6 51 36 61 26 71 16 41)(8 65 14 67 28 45 34 47)(10 79 32 73 30 59 12 53)(13 80 19 42 33 60 39 62)(15 54 37 48 35 74 17 68)

G:=sub<Sym(80)| (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)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80), (1,76,31,46,21,56,11,66)(2,63,40,49,22,43,20,69)(3,50,9,52,23,70,29,72)(4,77,18,55,24,57,38,75)(5,64,27,58,25,44,7,78)(6,51,36,61,26,71,16,41)(8,65,14,67,28,45,34,47)(10,79,32,73,30,59,12,53)(13,80,19,42,33,60,39,62)(15,54,37,48,35,74,17,68)>;

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)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80), (1,76,31,46,21,56,11,66)(2,63,40,49,22,43,20,69)(3,50,9,52,23,70,29,72)(4,77,18,55,24,57,38,75)(5,64,27,58,25,44,7,78)(6,51,36,61,26,71,16,41)(8,65,14,67,28,45,34,47)(10,79,32,73,30,59,12,53)(13,80,19,42,33,60,39,62)(15,54,37,48,35,74,17,68) );

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),(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80)], [(1,76,31,46,21,56,11,66),(2,63,40,49,22,43,20,69),(3,50,9,52,23,70,29,72),(4,77,18,55,24,57,38,75),(5,64,27,58,25,44,7,78),(6,51,36,61,26,71,16,41),(8,65,14,67,28,45,34,47),(10,79,32,73,30,59,12,53),(13,80,19,42,33,60,39,62),(15,54,37,48,35,74,17,68)])

C40.C4 is a maximal subgroup of
C804C4  C805C4  D40.C4  Dic20.C4  (C8×D5).C4  M4(2).1F5  D8⋊F5  SD163F5  Q16⋊F5  D10.Dic6  C40.Dic3
C40.C4 is a maximal quotient of
C402C8  Dic5.13D8  C20.10C42  D10.Dic6  C40.Dic3

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

0022
1011
2121
0021
,
0110
2021
1021
1111
G:=sub<GL(4,GF(3))| [0,1,2,0,0,0,1,0,2,1,2,2,2,1,1,1],[0,2,1,1,1,0,0,1,1,2,2,1,0,1,1,1] >;

C40.C4 in GAP, Magma, Sage, TeX

C_{40}.C_4
% in TeX

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

G:=SmallGroup(160,70);
// by ID

G=gap.SmallGroup(160,70);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,24,121,55,86,579,69,2309,1169]);
// Polycyclic

G:=Group<a,b|a^40=1,b^4=a^20,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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