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G = D40:C4  order 320 = 26·5

2nd semidirect product of D40 and C4 acting faithfully

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D8:3F5, D40:2C4, C8:2(C2xF5), C40:1(C2xC4), D4:D5:2C4, C5:(D8:C4), (C5xD8):2C4, D4:2(C2xF5), (D4xF5):2C2, D20:2(C2xC4), C40:C4:3C2, C8:F5:1C2, (D5xD8).3C2, (C2xF5).5D4, C2.16(D4xF5), D20:C4:2C2, C10.15(C4xD4), C4:F5.2C22, C4.2(C22xF5), D10.64(C2xD4), C20.2(C22xC4), D5:C8.1C22, (D4xD5).6C22, (C4xF5).1C22, D5.2(C8:C22), (C8xD5).13C22, (C4xD5).24C23, Dic5.2(C4oD4), (C5xD4):2(C2xC4), C5:2C8:12(C2xC4), SmallGroup(320,1069)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C20 — D40:C4
Chief series
C1 — C5 — C10 — D10 — C4xD5 — C4xF5 — D4xF5 — D40:C4
Lower central
C5 — C10 — C20 — D40:C4
Upper central
C1 — C2 — C4 — D8

Generators and relations for D40:C4
 G = < a,b,c | a40=b2=c4=1, bab=a-1, cac-1=a13, cbc-1=a32b >

Subgroups: 674 in 132 conjugacy classes, 42 normal (22 characteristic)
C1, C2, C2, C4, C4, C22, C5, C8, C8, C2xC4, D4, D4, C23, D5, D5, C10, C10, C42, C22:C4, C4:C4, C2xC8, D8, D8, C22xC4, C2xD4, Dic5, C20, F5, D10, D10, C2xC10, C8:C4, D4:C4, C4.Q8, C4xD4, C2xD8, C5:2C8, C40, C5:C8, C4xD5, D20, C5:D4, C5xD4, C2xF5, C2xF5, C22xD5, D8:C4, C8xD5, D40, D4:D5, C5xD8, D5:C8, C4xF5, C4:F5, C22:F5, D4xD5, C22xF5, C8:F5, C40:C4, D20:C4, D5xD8, D4xF5, D40:C4
Quotients: C1, C2, C4, C22, C2xC4, D4, C23, C22xC4, C2xD4, C4oD4, F5, C4xD4, C8:C22, C2xF5, D8:C4, C22xF5, D4xF5, D40:C4

Character table of D40:C4

 class 12A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J58A8B8C8D10A10B10C2040A40B
 size 114455202021010101010202020204420202041616888
ρ111111111111111111111111111111    trivial
ρ211-1-111-1-1111111-1-1-1-1111111-1-1111    linear of order 2
ρ3111-1111-111-1-1-1-1-111-11-111-111-11-1-1    linear of order 2
ρ411-1111-1111-1-1-1-11-1-111-111-11-111-1-1    linear of order 2
ρ51111111111-1-1-1-1-1-1-1-111-1-11111111    linear of order 2
ρ611-1-111-1-111-1-1-1-1111111-1-111-1-1111    linear of order 2
ρ7111-1111-11111111-1-111-1-1-1-111-11-1-1    linear of order 2
ρ811-1111-11111111-111-11-1-1-1-11-111-1-1    linear of order 2
ρ911-11-1-11-11-1ii-i-iii-i-i1-1i-i11-111-1-1    linear of order 4
ρ10111-1-1-1-111-1ii-i-i-i-iii1-1i-i111-11-1-1    linear of order 4
ρ11111-1-1-1-111-1-i-iiiii-i-i1-1-ii111-11-1-1    linear of order 4
ρ1211-11-1-11-11-1-i-iii-i-iii1-1-ii11-111-1-1    linear of order 4
ρ131111-1-1-1-11-1ii-i-i-ii-ii11-ii-1111111    linear of order 4
ρ1411-1-1-1-1111-1ii-i-ii-ii-i11-ii-11-1-1111    linear of order 4
ρ1511-1-1-1-1111-1-i-iii-ii-ii11i-i-11-1-1111    linear of order 4
ρ161111-1-1-1-11-1-i-iiii-ii-i11i-i-1111111    linear of order 4
ρ1722002200-2-2-22-22000020000200-200    orthogonal lifted from D4
ρ1822002200-2-22-22-2000020000200-200    orthogonal lifted from D4
ρ192200-2-200-22-2i2i2i-2i000020000200-200    complex lifted from C4oD4
ρ202200-2-200-222i-2i-2i2i000020000200-200    complex lifted from C4oD4
ρ2144-4-400004000000000-14000-111-1-1-1    orthogonal lifted from C2xF5
ρ224-400-4400000000000040000-400000    orthogonal lifted from C8:C22
ρ23444-400004000000000-1-4000-1-11-111    orthogonal lifted from C2xF5
ρ24444400004000000000-14000-1-1-1-1-1-1    orthogonal lifted from F5
ρ254-4004-400000000000040000-400000    orthogonal lifted from C8:C22
ρ2644-4400004000000000-1-4000-11-1-111    orthogonal lifted from C2xF5
ρ2788000000-8000000000-20000-200200    orthogonal lifted from D4xF5
ρ288-80000000000000000-200002000-√10√10    orthogonal faithful
ρ298-80000000000000000-200002000√10-√10    orthogonal faithful

Smallest permutation representation of D40: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)

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

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

Matrix representation of D40:C4 ►in GL8(F41)

250500000
41020000
3111600000
384031400000
00000001
000040404040
00001000
00000100
,
15000000
040000000
016100000
323140400000
000000040
000000400
000004000
000040000
,
400000000
040000000
100100000
41010000
00001000
00000001
00000100
000040404040

G:=sub<GL(8,GF(41))| [25,4,31,38,0,0,0,0,0,1,1,40,0,0,0,0,5,0,16,31,0,0,0,0,0,2,0,40,0,0,0,0,0,0,0,0,0,40,1,0,0,0,0,0,0,40,0,1,0,0,0,0,0,40,0,0,0,0,0,0,1,40,0,0],[1,0,0,32,0,0,0,0,5,40,16,31,0,0,0,0,0,0,1,40,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0],[40,0,10,4,0,0,0,0,0,40,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,40,0,0,0,0,0,0,1,40,0,0,0,0,0,0,0,40,0,0,0,0,0,1,0,40] >;
 

D40:C4 in GAP, Magma, Sage, TeX 

D_{40}\rtimes C_4
 
% in TeX
 
G:=Group("D40:C4");
 
// GroupNames label
 
G:=SmallGroup(320,1069);
 
// by ID
 
G=gap.SmallGroup(320,1069);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,758,219,851,438,102,6278,1595]);
 
// Polycyclic
 
G:=Group<a,b,c|a^40=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^13,c*b*c^-1=a^32*b>;
 
// generators/relations
 

Export

Character table of D40:C4 in TeX

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