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## G = C5×D4⋊2D5order 400 = 24·52

### Direct product of C5 and D4⋊2D5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C5×D4⋊2D5
 Chief series C1 — C5 — C10 — C5×C10 — D5×C10 — D5×C20 — C5×D4⋊2D5
 Lower central C5 — C10 — C5×D4⋊2D5
 Upper central C1 — C10 — C5×D4

Generators and relations for C5×D42D5
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, ece=b2c, ede=d-1 >

Subgroups: 236 in 96 conjugacy classes, 46 normal (26 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C5, C2×C4, D4, D4, Q8, D5, C10, C10, C4○D4, Dic5, Dic5, C20, C20, D10, C2×C10, C2×C10, C52, Dic10, C4×D5, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C5×D4, C5×Q8, C5×D5, C5×C10, C5×C10, D42D5, C5×C4○D4, C5×Dic5, C5×Dic5, C5×C20, D5×C10, C102, C5×Dic10, D5×C20, C10×Dic5, C5×C5⋊D4, D4×C52, C5×D42D5
Quotients: C1, C2, C22, C5, C23, D5, C10, C4○D4, D10, C2×C10, C22×D5, C22×C10, C5×D5, D42D5, C5×C4○D4, D5×C10, D5×C2×C10, C5×D42D5

Smallest permutation representation of C5×D42D5
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 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 13)(2 14)(3 15)(4 11)(5 12)(6 16)(7 17)(8 18)(9 19)(10 20)(21 36)(22 37)(23 38)(24 39)(25 40)(26 31)(27 32)(28 33)(29 34)(30 35)
(1 4 2 5 3)(6 9 7 10 8)(11 14 12 15 13)(16 19 17 20 18)(21 23 25 22 24)(26 28 30 27 29)(31 33 35 32 34)(36 38 40 37 39)
(1 21)(2 22)(3 23)(4 24)(5 25)(6 29)(7 30)(8 26)(9 27)(10 28)(11 34)(12 35)(13 31)(14 32)(15 33)(16 39)(17 40)(18 36)(19 37)(20 38)

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

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

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

100 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 5A 5B 5C 5D 5E ··· 5N 10A 10B 10C 10D 10E ··· 10V 10W ··· 10AP 10AQ 10AR 10AS 10AT 20A 20B 20C 20D 20E ··· 20N 20O ··· 20V 20W ··· 20AD order 1 2 2 2 2 4 4 4 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 10 2 5 5 10 10 1 1 1 1 2 ··· 2 1 1 1 1 2 ··· 2 4 ··· 4 10 10 10 10 2 2 2 2 4 ··· 4 5 ··· 5 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 D5 C4○D4 D10 D10 C5×D5 C5×C4○D4 D5×C10 D5×C10 D4⋊2D5 C5×D4⋊2D5 kernel C5×D4⋊2D5 C5×Dic10 D5×C20 C10×Dic5 C5×C5⋊D4 D4×C52 D4⋊2D5 Dic10 C4×D5 C2×Dic5 C5⋊D4 C5×D4 C5×D4 C52 C20 C2×C10 D4 C5 C4 C22 C5 C1 # reps 1 1 1 2 2 1 4 4 4 8 8 4 2 2 2 4 8 8 8 16 2 8

Matrix representation of C5×D42D5 in GL4(𝔽41) generated by

 16 0 0 0 0 16 0 0 0 0 37 0 0 0 0 37
,
 0 1 0 0 40 0 0 0 0 0 1 0 0 0 0 1
,
 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 18 0 0 0 19 16
,
 0 32 0 0 9 0 0 0 0 0 16 35 0 0 22 25
G:=sub<GL(4,GF(41))| [16,0,0,0,0,16,0,0,0,0,37,0,0,0,0,37],[0,40,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,18,19,0,0,0,16],[0,9,0,0,32,0,0,0,0,0,16,22,0,0,35,25] >;

C5×D42D5 in GAP, Magma, Sage, TeX

C_5\times D_4\rtimes_2D_5
% in TeX

G:=Group("C5xD4:2D5");
// GroupNames label

G:=SmallGroup(400,186);
// by ID

G=gap.SmallGroup(400,186);
# by ID

G:=PCGroup([6,-2,-2,-2,-5,-2,-5,247,794,404,11525]);
// 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,e*c*e=b^2*c,e*d*e=d^-1>;
// generators/relations

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