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## G = C5×D4⋊D5order 400 = 24·52

### Direct product of C5 and D4⋊D5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C5×D4⋊D5
 Chief series C1 — C5 — C10 — C20 — C5×C20 — C5×D20 — C5×D4⋊D5
 Lower central C5 — C10 — C20 — C5×D4⋊D5
 Upper central C1 — C10 — C20 — C5×D4

Generators and relations for C5×D4⋊D5
G = < a,b,c,d,e | a5=b4=c2=d5=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe=b-1, bd=db, cd=dc, ece=bc, ede=d-1 >

Subgroups: 188 in 60 conjugacy classes, 22 normal (all characteristic)
C1, C2, C2, C4, C22, C5, C5, C8, D4, D4, D5, C10, C10, D8, C20, C20, D10, C2×C10, C52, C52C8, C40, D20, C5×D4, C5×D4, C5×D5, C5×C10, C5×C10, D4⋊D5, C5×D8, C5×C20, D5×C10, C102, C5×C52C8, C5×D20, D4×C52, C5×D4⋊D5
Quotients: C1, C2, C22, C5, D4, D5, C10, D8, D10, C2×C10, C5⋊D4, C5×D4, C5×D5, D4⋊D5, C5×D8, D5×C10, C5×C5⋊D4, C5×D4⋊D5

Smallest permutation representation of C5×D4⋊D5
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 17 7 12)(2 18 8 13)(3 19 9 14)(4 20 10 15)(5 16 6 11)(21 31 26 36)(22 32 27 37)(23 33 28 38)(24 34 29 39)(25 35 30 40)
(1 12)(2 13)(3 14)(4 15)(5 11)(6 16)(7 17)(8 18)(9 19)(10 20)(21 26)(22 27)(23 28)(24 29)(25 30)
(1 3 5 2 4)(6 8 10 7 9)(11 13 15 12 14)(16 18 20 17 19)(21 24 22 25 23)(26 29 27 30 28)(31 34 32 35 33)(36 39 37 40 38)
(1 24)(2 25)(3 21)(4 22)(5 23)(6 28)(7 29)(8 30)(9 26)(10 27)(11 33)(12 34)(13 35)(14 31)(15 32)(16 38)(17 39)(18 40)(19 36)(20 37)

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

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

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

85 conjugacy classes

 class 1 2A 2B 2C 4 5A 5B 5C 5D 5E ··· 5N 8A 8B 10A 10B 10C 10D 10E ··· 10N 10O ··· 10AL 10AM 10AN 10AO 10AP 20A 20B 20C 20D 20E ··· 20N 40A ··· 40H order 1 2 2 2 4 5 5 5 5 5 ··· 5 8 8 10 10 10 10 10 ··· 10 10 ··· 10 10 10 10 10 20 20 20 20 20 ··· 20 40 ··· 40 size 1 1 4 20 2 1 1 1 1 2 ··· 2 10 10 1 1 1 1 2 ··· 2 4 ··· 4 20 20 20 20 2 2 2 2 4 ··· 4 10 ··· 10

85 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + image C1 C2 C2 C2 C5 C10 C10 C10 D4 D5 D8 D10 C5⋊D4 C5×D4 C5×D5 C5×D8 D5×C10 C5×C5⋊D4 D4⋊D5 C5×D4⋊D5 kernel C5×D4⋊D5 C5×C5⋊2C8 C5×D20 D4×C52 D4⋊D5 C5⋊2C8 D20 C5×D4 C5×C10 C5×D4 C52 C20 C10 C10 D4 C5 C4 C2 C5 C1 # reps 1 1 1 1 4 4 4 4 1 2 2 2 4 4 8 8 8 16 2 8

Matrix representation of C5×D4⋊D5 in GL4(𝔽41) generated by

 37 0 0 0 0 37 0 0 0 0 1 0 0 0 0 1
,
 40 0 0 0 0 40 0 0 0 0 1 2 0 0 40 40
,
 40 0 0 0 0 1 0 0 0 0 40 0 0 0 1 1
,
 16 0 0 0 0 18 0 0 0 0 1 0 0 0 0 1
,
 0 18 0 0 16 0 0 0 0 0 24 24 0 0 29 17
G:=sub<GL(4,GF(41))| [37,0,0,0,0,37,0,0,0,0,1,0,0,0,0,1],[40,0,0,0,0,40,0,0,0,0,1,40,0,0,2,40],[40,0,0,0,0,1,0,0,0,0,40,1,0,0,0,1],[16,0,0,0,0,18,0,0,0,0,1,0,0,0,0,1],[0,16,0,0,18,0,0,0,0,0,24,29,0,0,24,17] >;

C5×D4⋊D5 in GAP, Magma, Sage, TeX

C_5\times D_4\rtimes D_5
% in TeX

G:=Group("C5xD4:D5");
// GroupNames label

G:=SmallGroup(400,87);
// by ID

G=gap.SmallGroup(400,87);
# by ID

G:=PCGroup([6,-2,-2,-5,-2,-2,-5,265,1443,729,69,11525]);
// Polycyclic

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

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