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## G = C4×D52order 400 = 24·52

### Direct product of C4, D5 and D5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — C4×D52
 Chief series C1 — C5 — C52 — C5×C10 — D5×C10 — C2×D52 — C4×D52
 Lower central C52 — C4×D52
 Upper central C1 — C4

Generators and relations for C4×D52
G = < a,b,c,d,e | a4=b5=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: 748 in 124 conjugacy classes, 46 normal (12 characteristic)
C1, C2, C2, C4, C4, C22, C5, C5, C2×C4, C23, D5, D5, C10, C10, C22×C4, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C52, C4×D5, C4×D5, C2×Dic5, C2×C20, C22×D5, C5×D5, C5⋊D5, C5×C10, C2×C4×D5, C5×Dic5, C526C4, C5×C20, D52, D5×C10, C2×C5⋊D5, D5×Dic5, Dic52D5, D5×C20, C4×C5⋊D5, C2×D52, C4×D52
Quotients: C1, C2, C4, C22, C2×C4, C23, D5, C22×C4, D10, C4×D5, C22×D5, C2×C4×D5, D52, C2×D52, C4×D52

Smallest permutation representation of C4×D52
On 40 points
Generators in S40
(1 18 8 13)(2 19 9 14)(3 20 10 15)(4 16 6 11)(5 17 7 12)(21 36 26 31)(22 37 27 32)(23 38 28 33)(24 39 29 34)(25 40 30 35)
(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 21)(2 25)(3 24)(4 23)(5 22)(6 28)(7 27)(8 26)(9 30)(10 29)(11 33)(12 32)(13 31)(14 35)(15 34)(16 38)(17 37)(18 36)(19 40)(20 39)
(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 29)(2 30)(3 26)(4 27)(5 28)(6 22)(7 23)(8 24)(9 25)(10 21)(11 37)(12 38)(13 39)(14 40)(15 36)(16 32)(17 33)(18 34)(19 35)(20 31)

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

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

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

64 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 4G 4H 5A 5B 5C 5D 5E 5F 5G 5H 10A 10B 10C 10D 10E 10F 10G 10H 10I ··· 10P 20A ··· 20H 20I ··· 20P 20Q ··· 20X order 1 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 10 10 10 10 10 10 10 10 10 ··· 10 20 ··· 20 20 ··· 20 20 ··· 20 size 1 1 5 5 5 5 25 25 1 1 5 5 5 5 25 25 2 2 2 2 4 4 4 4 2 2 2 2 4 4 4 4 10 ··· 10 2 ··· 2 4 ··· 4 10 ··· 10

64 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 D5 D10 D10 D10 C4×D5 D52 C2×D52 C4×D52 kernel C4×D52 D5×Dic5 Dic5⋊2D5 D5×C20 C4×C5⋊D5 C2×D52 D52 C4×D5 Dic5 C20 D10 D5 C4 C2 C1 # reps 1 2 1 2 1 1 8 4 4 4 4 16 4 4 8

Matrix representation of C4×D52 in GL4(𝔽41) generated by

 9 0 0 0 0 9 0 0 0 0 1 0 0 0 0 1
,
 6 40 0 0 1 0 0 0 0 0 1 0 0 0 0 1
,
 6 40 0 0 35 35 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 6 40 0 0 1 0
,
 40 0 0 0 0 40 0 0 0 0 6 40 0 0 35 35
G:=sub<GL(4,GF(41))| [9,0,0,0,0,9,0,0,0,0,1,0,0,0,0,1],[6,1,0,0,40,0,0,0,0,0,1,0,0,0,0,1],[6,35,0,0,40,35,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,6,1,0,0,40,0],[40,0,0,0,0,40,0,0,0,0,6,35,0,0,40,35] >;

C4×D52 in GAP, Magma, Sage, TeX

C_4\times D_5^2
% in TeX

G:=Group("C4xD5^2");
// GroupNames label

G:=SmallGroup(400,169);
// by ID

G=gap.SmallGroup(400,169);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-5,-5,50,970,11525]);
// Polycyclic

G:=Group<a,b,c,d,e|a^4=b^5=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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