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G = Dic5.F5order 400 = 24·52

1st non-split extension by Dic5 of F5 acting via F5/D5=C2

metabelian, supersoluble, monomial

Aliases: Dic5.1F5, C523M4(2), Dic5.8D10, C5⋊C82D5, C2.7(D5×F5), C10.7(C4×D5), C54(C4.F5), C51(C8⋊D5), C523C84C2, C10.34(C2×F5), (C5×Dic5).1C4, Dic52D5.3C2, (C5×Dic5).9C22, (C5×C5⋊C8)⋊4C2, (C2×C5⋊D5).2C4, (C5×C10).7(C2×C4), SmallGroup(400,123)

Series: Derived Chief Lower central Upper central

C1C5×C10 — Dic5.F5
C1C5C52C5×C10C5×Dic5C5×C5⋊C8 — Dic5.F5
C52C5×C10 — Dic5.F5
C1C2

Generators and relations for Dic5.F5
 G = < a,b,c,d | a10=c5=1, b2=d4=a5, bab-1=a-1, ac=ca, ad=da, bc=cb, dbd-1=a5b, dcd-1=c3 >

50C2
4C5
5C4
5C4
25C22
4C10
10D5
10D5
20D5
20D5
5C8
25C2×C4
25C8
5D10
5C20
5D10
5C20
20D10
2C5⋊D5
25M4(2)
5C4×D5
5C4×D5
5C5⋊C8
5C40
5C52C8
5C8⋊D5
5C4.F5

Smallest permutation representation of Dic5.F5
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 11 6 16)(2 20 7 15)(3 19 8 14)(4 18 9 13)(5 17 10 12)(21 36 26 31)(22 35 27 40)(23 34 28 39)(24 33 29 38)(25 32 30 37)
(1 7 3 9 5)(2 8 4 10 6)(11 15 19 13 17)(12 16 20 14 18)(21 23 25 27 29)(22 24 26 28 30)(31 39 37 35 33)(32 40 38 36 34)
(1 36 16 26 6 31 11 21)(2 37 17 27 7 32 12 22)(3 38 18 28 8 33 13 23)(4 39 19 29 9 34 14 24)(5 40 20 30 10 35 15 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), (1,11,6,16)(2,20,7,15)(3,19,8,14)(4,18,9,13)(5,17,10,12)(21,36,26,31)(22,35,27,40)(23,34,28,39)(24,33,29,38)(25,32,30,37), (1,7,3,9,5)(2,8,4,10,6)(11,15,19,13,17)(12,16,20,14,18)(21,23,25,27,29)(22,24,26,28,30)(31,39,37,35,33)(32,40,38,36,34), (1,36,16,26,6,31,11,21)(2,37,17,27,7,32,12,22)(3,38,18,28,8,33,13,23)(4,39,19,29,9,34,14,24)(5,40,20,30,10,35,15,25)>;

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

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

34 conjugacy classes

class 1 2A2B4A4B4C5A5B5C5D5E8A8B8C8D10A10B10C10D10E20A20B20C20D20E20F40A···40H
order122444555558888101010101020202020202040···40
size1150551022488101050502248810101010202010···10

34 irreducible representations

dim1111112222244488
type++++++++++
imageC1C2C2C2C4C4D5M4(2)D10C4×D5C8⋊D5F5C2×F5C4.F5D5×F5Dic5.F5
kernelDic5.F5C5×C5⋊C8C523C8Dic52D5C5×Dic5C2×C5⋊D5C5⋊C8C52Dic5C10C5Dic5C10C5C2C1
# reps1111222224811222

Matrix representation of Dic5.F5 in GL6(𝔽41)

3510000
4000000
001000
000100
000010
000001
,
3200000
2890000
001000
000100
000010
000001
,
100000
010000
0040100
0040001
0040000
0040010
,
13230000
18280000
000001
001000
000100
000010

G:=sub<GL(6,GF(41))| [35,40,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[32,28,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,40,40,40,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0],[13,18,0,0,0,0,23,28,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0] >;

Dic5.F5 in GAP, Magma, Sage, TeX

{\rm Dic}_5.F_5
% in TeX

G:=Group("Dic5.F5");
// GroupNames label

G:=SmallGroup(400,123);
// by ID

G=gap.SmallGroup(400,123);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-5,-5,24,121,55,50,970,5765,2897]);
// Polycyclic

G:=Group<a,b,c,d|a^10=c^5=1,b^2=d^4=a^5,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=a^5*b,d*c*d^-1=c^3>;
// generators/relations

Export

Subgroup lattice of Dic5.F5 in TeX

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