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## G = C5×D4×D5order 400 = 24·52

### Direct product of C5, D4 and D5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C5×D4×D5
 Chief series C1 — C5 — C10 — C5×C10 — D5×C10 — D5×C2×C10 — C5×D4×D5
 Lower central C5 — C10 — C5×D4×D5
 Upper central C1 — C10 — 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=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 380 in 124 conjugacy classes, 50 normal (26 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C5, C2×C4, D4, D4, C23, D5, D5, C10, C10, C2×D4, Dic5, C20, C20, D10, D10, D10, C2×C10, C2×C10, C52, C4×D5, D20, C5⋊D4, C2×C20, C5×D4, C5×D4, C22×D5, C22×C10, C5×D5, C5×D5, C5×C10, C5×C10, D4×D5, D4×C10, C5×Dic5, C5×C20, D5×C10, D5×C10, D5×C10, C102, D5×C20, C5×D20, C5×C5⋊D4, D4×C52, D5×C2×C10, C5×D4×D5
Quotients: C1, C2, C22, C5, D4, C23, D5, C10, C2×D4, D10, C2×C10, C5×D4, C22×D5, C22×C10, C5×D5, D4×D5, D4×C10, D5×C10, D5×C2×C10, 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 11 6 16)(2 12 7 17)(3 13 8 18)(4 14 9 19)(5 15 10 20)(21 31 26 36)(22 32 27 37)(23 33 28 38)(24 34 29 39)(25 35 30 40)
(11 16)(12 17)(13 18)(14 19)(15 20)(31 36)(32 37)(33 38)(34 39)(35 40)
(1 5 4 3 2)(6 10 9 8 7)(11 15 14 13 12)(16 20 19 18 17)(21 22 23 24 25)(26 27 28 29 30)(31 32 33 34 35)(36 37 38 39 40)
(1 25)(2 21)(3 22)(4 23)(5 24)(6 30)(7 26)(8 27)(9 28)(10 29)(11 35)(12 31)(13 32)(14 33)(15 34)(16 40)(17 36)(18 37)(19 38)(20 39)

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

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

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

100 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 5A 5B 5C 5D 5E ··· 5N 10A 10B 10C 10D 10E ··· 10V 10W ··· 10AP 10AQ ··· 10AX 10AY ··· 10BF 20A 20B 20C 20D 20E ··· 20N 20O 20P 20Q 20R order 1 2 2 2 2 2 2 2 4 4 5 5 5 5 5 ··· 5 10 10 10 10 10 ··· 10 10 ··· 10 10 ··· 10 10 ··· 10 20 20 20 20 20 ··· 20 20 20 20 20 size 1 1 2 2 5 5 10 10 2 10 1 1 1 1 2 ··· 2 1 1 1 1 2 ··· 2 4 ··· 4 5 ··· 5 10 ··· 10 2 2 2 2 4 ··· 4 10 10 10 10

100 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C5 C10 C10 C10 C10 C10 D4 D5 D10 D10 C5×D4 C5×D5 D5×C10 D5×C10 D4×D5 C5×D4×D5 kernel C5×D4×D5 D5×C20 C5×D20 C5×C5⋊D4 D4×C52 D5×C2×C10 D4×D5 C4×D5 D20 C5⋊D4 C5×D4 C22×D5 C5×D5 C5×D4 C20 C2×C10 D5 D4 C4 C22 C5 C1 # reps 1 1 1 2 1 2 4 4 4 8 4 8 2 2 2 4 8 8 8 16 2 8

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

 18 0 0 0 0 18 0 0 0 0 1 0 0 0 0 1
,
 40 0 0 0 0 40 0 0 0 0 0 40 0 0 1 0
,
 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 40
,
 10 0 0 0 0 37 0 0 0 0 1 0 0 0 0 1
,
 0 37 0 0 10 0 0 0 0 0 40 0 0 0 0 40
G:=sub<GL(4,GF(41))| [18,0,0,0,0,18,0,0,0,0,1,0,0,0,0,1],[40,0,0,0,0,40,0,0,0,0,0,1,0,0,40,0],[1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,40],[10,0,0,0,0,37,0,0,0,0,1,0,0,0,0,1],[0,10,0,0,37,0,0,0,0,0,40,0,0,0,0,40] >;

C5×D4×D5 in GAP, Magma, Sage, TeX

C_5\times D_4\times D_5
% in TeX

G:=Group("C5xD4xD5");
// GroupNames label

G:=SmallGroup(400,185);
// by ID

G=gap.SmallGroup(400,185);
# by ID

G:=PCGroup([6,-2,-2,-2,-5,-2,-5,404,11525]);
// 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=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations

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