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G = C20.31D10order 400 = 24·52

5th non-split extension by C20 of D10 acting via D10/D5=C2

metabelian, supersoluble, monomial

Aliases: C20.31D10, C52:10M4(2), C4.16D52, C5:2C8:5D5, C10.9(C4xD5), C5:3(C8:D5), C52:6C4.5C4, (C5xC20).30C22, C2.3(Dic5:2D5), (C5xC5:2C8):7C2, (C4xC5:D5).4C2, (C2xC5:D5).5C4, (C5xC10).44(C2xC4), SmallGroup(400,63)

Series: Derived Chief Lower central Upper central

C1C5xC10 — C20.31D10
C1C5C52C5xC10C5xC20C5xC5:2C8 — C20.31D10
C52C5xC10 — C20.31D10
C1C4

Generators and relations for C20.31D10
 G = < a,b,c | a20=c2=1, b10=a5, bab-1=cac=a9, cbc=a10b9 >

Subgroups: 364 in 56 conjugacy classes, 20 normal (10 characteristic)
C1, C2, C2, C4, C4, C22, C5, C5, C8, C2xC4, D5, C10, C10, M4(2), Dic5, C20, C20, D10, C52, C5:2C8, C40, C4xD5, C5:D5, C5xC10, C8:D5, C52:6C4, C5xC20, C2xC5:D5, C5xC5:2C8, C4xC5:D5, C20.31D10
Quotients: C1, C2, C4, C22, C2xC4, D5, M4(2), D10, C4xD5, C8:D5, D52, Dic5:2D5, C20.31D10

Smallest permutation representation of C20.31D10
On 40 points
Generators in S40
(1 35 29 23 17 11 5 39 33 27 21 15 9 3 37 31 25 19 13 7)(2 28 14 40 26 12 38 24 10 36 22 8 34 20 6 32 18 4 30 16)
(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 5)(2 34)(3 23)(4 12)(6 30)(7 19)(9 37)(10 26)(11 15)(13 33)(14 22)(16 40)(17 29)(20 36)(21 25)(24 32)(27 39)(31 35)

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

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

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

58 conjugacy classes

class 1 2A2B4A4B4C5A5B5C5D5E5F5G5H8A8B8C8D10A10B10C10D10E10F10G10H20A···20H20I···20P40A···40P
order122444555555558888101010101010101020···2020···2040···40
size115011502222444410101010222244442···24···410···10

58 irreducible representations

dim1111122222444
type+++++++
imageC1C2C2C4C4D5M4(2)D10C4xD5C8:D5D52Dic5:2D5C20.31D10
kernelC20.31D10C5xC5:2C8C4xC5:D5C52:6C4C2xC5:D5C5:2C8C52C20C10C5C4C2C1
# reps12122424816448

Matrix representation of C20.31D10 in GL4(F41) generated by

9000
0900
0001
004034
,
31900
32900
0010
003440
,
1700
04000
0010
003440
G:=sub<GL(4,GF(41))| [9,0,0,0,0,9,0,0,0,0,0,40,0,0,1,34],[31,32,0,0,9,9,0,0,0,0,1,34,0,0,0,40],[1,0,0,0,7,40,0,0,0,0,1,34,0,0,0,40] >;

C20.31D10 in GAP, Magma, Sage, TeX

C_{20}._{31}D_{10}
% in TeX

G:=Group("C20.31D10");
// GroupNames label

G:=SmallGroup(400,63);
// by ID

G=gap.SmallGroup(400,63);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-5,-5,24,121,31,50,970,11525]);
// Polycyclic

G:=Group<a,b,c|a^20=c^2=1,b^10=a^5,b*a*b^-1=c*a*c=a^9,c*b*c=a^10*b^9>;
// generators/relations

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