Copied to
clipboard

## G = C5×C22.F5order 400 = 24·52

### Direct product of C5 and C22.F5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C5×C22.F5
 Chief series C1 — C5 — C10 — Dic5 — C5×Dic5 — C5×C5⋊C8 — C5×C22.F5
 Lower central C5 — C10 — C5×C22.F5
 Upper central C1 — C10 — C2×C10

Generators and relations for C5×C22.F5
G = < a,b,c,d,e | a5=b2=c2=d5=1, e4=c, ab=ba, ac=ca, ad=da, ae=ea, ebe-1=bc=cb, bd=db, cd=dc, ce=ec, ede-1=d3 >

Smallest permutation representation of C5×C22.F5
On 40 points
Generators in S40
(1 22 35 26 9)(2 23 36 27 10)(3 24 37 28 11)(4 17 38 29 12)(5 18 39 30 13)(6 19 40 31 14)(7 20 33 32 15)(8 21 34 25 16)
(2 6)(4 8)(10 14)(12 16)(17 21)(19 23)(25 29)(27 31)(34 38)(36 40)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)(33 37)(34 38)(35 39)(36 40)
(1 35 9 22 26)(2 23 36 27 10)(3 28 24 11 37)(4 12 29 38 17)(5 39 13 18 30)(6 19 40 31 14)(7 32 20 15 33)(8 16 25 34 21)
(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)

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

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

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

70 conjugacy classes

 class 1 2A 2B 4A 4B 4C 5A 5B 5C 5D 5E ··· 5I 8A 8B 8C 8D 10A 10B 10C 10D 10E 10F 10G 10H 10I ··· 10W 20A ··· 20H 20I 20J 20K 20L 40A ··· 40P order 1 2 2 4 4 4 5 5 5 5 5 ··· 5 8 8 8 8 10 10 10 10 10 10 10 10 10 ··· 10 20 ··· 20 20 20 20 20 40 ··· 40 size 1 1 2 5 5 10 1 1 1 1 4 ··· 4 10 10 10 10 1 1 1 1 2 2 2 2 4 ··· 4 5 ··· 5 10 10 10 10 10 ··· 10

70 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 4 4 4 4 4 4 type + + + + + - image C1 C2 C2 C4 C4 C5 C10 C10 C20 C20 M4(2) C5×M4(2) F5 C2×F5 C22.F5 C5×F5 C10×F5 C5×C22.F5 kernel C5×C22.F5 C5×C5⋊C8 C10×Dic5 C5×Dic5 C102 C22.F5 C5⋊C8 C2×Dic5 Dic5 C2×C10 C52 C5 C2×C10 C10 C5 C22 C2 C1 # reps 1 2 1 2 2 4 8 4 8 8 2 8 1 1 2 4 4 8

Matrix representation of C5×C22.F5 in GL4(𝔽41) generated by

 37 0 0 0 0 37 0 0 0 0 37 0 0 0 0 37
,
 1 0 0 0 0 1 0 0 0 0 40 0 0 0 0 40
,
 40 0 0 0 0 40 0 0 0 0 40 0 0 0 0 40
,
 16 0 0 0 7 18 0 0 0 0 10 0 0 0 33 37
,
 0 0 1 0 0 0 0 1 32 15 0 0 0 9 0 0
G:=sub<GL(4,GF(41))| [37,0,0,0,0,37,0,0,0,0,37,0,0,0,0,37],[1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[16,7,0,0,0,18,0,0,0,0,10,33,0,0,0,37],[0,0,32,0,0,0,15,9,1,0,0,0,0,1,0,0] >;

C5×C22.F5 in GAP, Magma, Sage, TeX

C_5\times C_2^2.F_5
% in TeX

G:=Group("C5xC2^2.F5");
// GroupNames label

G:=SmallGroup(400,140);
// by ID

G=gap.SmallGroup(400,140);
# by ID

G:=PCGroup([6,-2,-2,-5,-2,-2,-5,120,505,69,5765,599]);
// Polycyclic

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

Export

׿
×
𝔽