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## G = C2×D5×F5order 400 = 24·52

### Direct product of C2, D5 and F5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — C2×D5×F5
 Chief series C1 — C5 — C52 — C5×D5 — C5×F5 — D5×F5 — C2×D5×F5
 Lower central C52 — C2×D5×F5
 Upper central C1 — C2

Generators and relations for C2×D5×F5
G = < a,b,c,d,e | a2=b5=c2=d5=e4=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d3 >

Subgroups: 760 in 113 conjugacy classes, 42 normal (22 characteristic)
C1, C2, C2, C4, C22, C5, C5, C2×C4, C23, D5, D5, D5, C10, C10, C22×C4, Dic5, C20, F5, F5, D10, D10, C2×C10, C52, C4×D5, C2×Dic5, C2×C20, C2×F5, C2×F5, C22×D5, C5×D5, C5×D5, C5⋊D5, C5×C10, C2×C4×D5, C22×F5, C5×F5, D5.D5, D52, D5×C10, C2×C5⋊D5, D5×F5, C10×F5, C2×D5.D5, C2×D52, C2×D5×F5
Quotients: C1, C2, C4, C22, C2×C4, C23, D5, C22×C4, F5, D10, C4×D5, C2×F5, C22×D5, C2×C4×D5, C22×F5, D5×F5, C2×D5×F5

Smallest permutation representation of C2×D5×F5
On 40 points
Generators in S40
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 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 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 12)(10 11)(21 40)(22 39)(23 38)(24 37)(25 36)(26 35)(27 34)(28 33)(29 32)(30 31)
(1 2 3 4 5)(6 10 9 8 7)(11 12 13 14 15)(16 20 19 18 17)(21 23 25 22 24)(26 29 27 30 28)(31 33 35 32 34)(36 39 37 40 38)
(1 36 6 31)(2 37 7 32)(3 38 8 33)(4 39 9 34)(5 40 10 35)(11 26 16 21)(12 27 17 22)(13 28 18 23)(14 29 19 24)(15 30 20 25)

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

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

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

40 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 4G 4H 5A 5B 5C 5D 5E 10A 10B 10C 10D 10E 10F 10G 10H 10I 10J 10K 20A ··· 20H order 1 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 5 5 5 5 5 10 10 10 10 10 10 10 10 10 10 10 20 ··· 20 size 1 1 5 5 5 5 25 25 5 5 5 5 25 25 25 25 2 2 4 8 8 2 2 4 8 8 10 10 10 10 20 20 10 ··· 10

40 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 4 4 4 8 8 type + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C4 C4 C4 D5 D10 D10 C4×D5 C4×D5 F5 C2×F5 C2×F5 D5×F5 C2×D5×F5 kernel C2×D5×F5 D5×F5 C10×F5 C2×D5.D5 C2×D52 D52 D5×C10 C2×C5⋊D5 C2×F5 F5 D10 D5 C10 D10 D5 C10 C2 C1 # reps 1 4 1 1 1 4 2 2 2 4 2 4 4 1 2 1 2 2

Matrix representation of C2×D5×F5 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 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
,
 34 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
,
 7 1 0 0 0 0 34 34 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 0 1 0 0 0 0 0 0 1 0 0 0 40 40 40 40 0 0 1 0 0 0
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 40 40 40 40 0 0 0 1 0 0 0 0 0 0 1 0

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[34,1,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[7,34,0,0,0,0,1,34,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,1,0,0,1,0,40,0,0,0,0,1,40,0,0,0,0,0,40,0],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,40,0,0,0,0,0,40,1,0,0,0,0,40,0,1,0,0,0,40,0,0] >;

C2×D5×F5 in GAP, Magma, Sage, TeX

C_2\times D_5\times F_5
% in TeX

G:=Group("C2xD5xF5");
// GroupNames label

G:=SmallGroup(400,209);
// by ID

G=gap.SmallGroup(400,209);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-5,-5,48,970,5765,1463]);
// Polycyclic

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

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