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G = C20.11F5  order 400 = 24·52

11st non-split extension by C20 of F5 acting via F5/C5=C4

metabelian, supersoluble, monomial, A-group

Aliases: C20.11F5, C5⋊D5⋊5C8, C52⋊8(C2×C8), C5⋊2(D5⋊C8), (C5×C20).12C4, C52⋊5C8⋊7C2, C10.21(C2×F5), C4.3(C52⋊C4), C52⋊6C4.22C22, (C2×C5⋊D5).9C4, (C4×C5⋊D5).12C2, C2.1(C2×C52⋊C4), (C5×C10).34(C2×C4), SmallGroup(400,156)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C52 — C20.11F5
Chief series
C1 — C5 — C52 — C5×C10 — C52⋊6C4 — C52⋊5C8 — C20.11F5
Lower central
C52 — C20.11F5
Upper central
C1 — C4

Generators and relations for C20.11F5
 G = < a,b,c | a20=b5=1, c4=a10, ab=ba, cac-1=a17, cbc-1=b3 >

Subgroups: 412 in 60 conjugacy classes, 20 normal (10 characteristic)
C1, C2, C2, C4, C4, C22, C5, C5, C8, C2×C4, D5, C10, C10, C2×C8, Dic5, C20, C20, D10, C52, C5⋊C8, C4×D5, C5⋊D5, C5×C10, D5⋊C8, C52⋊6C4, C5×C20, C2×C5⋊D5, C52⋊5C8, C4×C5⋊D5, C20.11F5
Quotients: C1, C2, C4, C22, C8, C2×C4, C2×C8, F5, C2×F5, D5⋊C8, C52⋊C4, C2×C52⋊C4, C20.11F5

Smallest permutation representation of C20.11F5
►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 9 17 5 13)(2 10 18 6 14)(3 11 19 7 15)(4 12 20 8 16)(21 33 25 37 29)(22 34 26 38 30)(23 35 27 39 31)(24 36 28 40 32)

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

40 conjugacy classes

class 1 2A2B2C4A4B4C4D5A···5F8A···8H10A···10F20A···20L
order122244445···58···810···1020···20
size1125251125254···425···254···44···4

40 irreducible representations

dim111111444444
type+++++++
imageC1C2C2C4C4C8F5C2×F5D5⋊C8C52⋊C4C2×C52⋊C4C20.11F5
kernelC20.11F5C52⋊5C8C4×C5⋊D5C5×C20C2×C5⋊D5C5⋊D5C20C10C5C4C2C1
# reps121228224448

Matrix representation of C20.11F5 ►in GL5(𝔽41)

320000
0344000
01000
00077
0003440
,
10000
0403400
07700
00001
0004034
,
30000
00010
0003440
01000
00100

G:=sub<GL(5,GF(41))| [32,0,0,0,0,0,34,1,0,0,0,40,0,0,0,0,0,0,7,34,0,0,0,7,40],[1,0,0,0,0,0,40,7,0,0,0,34,7,0,0,0,0,0,0,40,0,0,0,1,34],[3,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,1,34,0,0,0,0,40,0,0] >;
 

C20.11F5 in GAP, Magma, Sage, TeX 

C_{20}._{11}F_5
 
% in TeX
 
G:=Group("C20.11F5");
 
// GroupNames label
 
G:=SmallGroup(400,156);
 
// by ID
 
G=gap.SmallGroup(400,156);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-2,-5,-5,24,55,50,1444,496,5765,2897]);
 
// Polycyclic
 
G:=Group<a,b,c|a^20=b^5=1,c^4=a^10,a*b=b*a,c*a*c^-1=a^17,c*b*c^-1=b^3>;
 
// generators/relations
 

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