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G = C5⋊D40  order 400 = 24·52

The semidirect product of C5 and D40 acting via D40/D20=C2

metabelian, supersoluble, monomial

Aliases: C5⋊2D40, C52⋊3D8, D20⋊1D5, C20.10D10, C10.12D20, C4.1D52, C5⋊2C8⋊1D5, C5⋊1(D4⋊D5), (C5×D20)⋊2C2, (C5×C10).7D4, C20⋊D5⋊2C2, C10.1(C5⋊D4), (C5×C20).2C22, C2.4(C5⋊D20), (C5×C5⋊2C8)⋊1C2, SmallGroup(400,65)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C5×C20 — C5⋊D40
Chief series
C1 — C5 — C52 — C5×C10 — C5×C20 — C5×D20 — C5⋊D40
Lower central
C52 — C5×C10 — C5×C20 — C5⋊D40
Upper central
C1 — C2 — C4

Generators and relations for C5⋊D40
 G = < a,b,c | a5=b40=c2=1, bab-1=cac=a-1, cbc=b-1 >

Subgroups: 548 in 60 conjugacy classes, 18 normal (all characteristic)
C1, C2, C2, C4, C22, C5, C5, C8, D4, D5, C10, C10, D8, C20, C20, D10, C2×C10, C52, C5⋊2C8, C40, D20, D20, C5×D4, C5×D5, C5⋊D5, C5×C10, D40, D4⋊D5, C5×C20, D5×C10, C2×C5⋊D5, C5×C5⋊2C8, C5×D20, C20⋊D5, C5⋊D40
Quotients: C1, C2, C22, D4, D5, D8, D10, D20, C5⋊D4, D40, D4⋊D5, D52, C5⋊D20, C5⋊D40

Smallest permutation representation of C5⋊D40
►On 40 points
Generators in S40
(1 25 9 33 17)(2 18 34 10 26)(3 27 11 35 19)(4 20 36 12 28)(5 29 13 37 21)(6 22 38 14 30)(7 31 15 39 23)(8 24 40 16 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 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)
 
G:=sub<Sym(40)| (1,25,9,33,17)(2,18,34,10,26)(3,27,11,35,19)(4,20,36,12,28)(5,29,13,37,21)(6,22,38,14,30)(7,31,15,39,23)(8,24,40,16,32), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40), (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)>;
 
G:=Group( (1,25,9,33,17)(2,18,34,10,26)(3,27,11,35,19)(4,20,36,12,28)(5,29,13,37,21)(6,22,38,14,30)(7,31,15,39,23)(8,24,40,16,32), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40), (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) );
 
G=PermutationGroup([[(1,25,9,33,17),(2,18,34,10,26),(3,27,11,35,19),(4,20,36,12,28),(5,29,13,37,21),(6,22,38,14,30),(7,31,15,39,23),(8,24,40,16,32)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40)], [(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)]])
 

49 conjugacy classes

class 1 2A2B2C 4 5A5B5C5D5E5F5G5H8A8B10A10B10C10D10E10F10G10H10I10J10K10L20A20B20C20D20E···20N40A···40H
order1222455555555881010101010101010101010102020202020···2040···40
size11201002222244441010222244442020202022224···410···10

49 irreducible representations

dim1111222222224444
type+++++++++++++++
imageC1C2C2C2D4D5D5D8D10D20C5⋊D4D40D4⋊D5D52C5⋊D20C5⋊D40
kernelC5⋊D40C5×C5⋊2C8C5×D20C20⋊D5C5×C10C5⋊2C8D20C52C20C10C10C5C5C4C2C1
# reps1111122244482448

Matrix representation of C5⋊D40 ►in GL6(𝔽41)

100000
010000
001000
000100
0000040
000016
,
24150000
3000000
00271100
00303200
00003523
0000186
,
100000
23400000
00344000
007700
0000040
0000400

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,40,6],[24,30,0,0,0,0,15,0,0,0,0,0,0,0,27,30,0,0,0,0,11,32,0,0,0,0,0,0,35,18,0,0,0,0,23,6],[1,23,0,0,0,0,0,40,0,0,0,0,0,0,34,7,0,0,0,0,40,7,0,0,0,0,0,0,0,40,0,0,0,0,40,0] >;
 

C5⋊D40 in GAP, Magma, Sage, TeX 

C_5\rtimes D_{40}
 
% in TeX
 
G:=Group("C5:D40");
 
// GroupNames label
 
G:=SmallGroup(400,65);
 
// by ID
 
G=gap.SmallGroup(400,65);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-2,-5,-5,73,79,218,50,970,11525]);
 
// Polycyclic
 
G:=Group<a,b,c|a^5=b^40=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;
 
// generators/relations
 

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