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## G = D5×C5⋊D4order 400 = 24·52

### Direct product of D5 and C5⋊D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5×C10 — D5×C5⋊D4
 Chief series C1 — C5 — C52 — C5×C10 — D5×C10 — C2×D52 — D5×C5⋊D4
 Lower central C52 — C5×C10 — D5×C5⋊D4
 Upper central C1 — C2 — C22

Generators and relations for D5×C5⋊D4
G = < a,b,c,d,e | a5=b2=c5=d4=e2=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=ece=c-1, ede=d-1 >

Subgroups: 836 in 124 conjugacy classes, 36 normal (32 characteristic)
C1, C2, C2, C4, C22, C22, C5, C5, C2×C4, D4, C23, D5, D5, C10, C10, C2×D4, Dic5, Dic5, C20, D10, D10, C2×C10, C2×C10, C52, C4×D5, D20, C2×Dic5, C5⋊D4, C5⋊D4, C5×D4, C22×D5, C22×D5, C22×C10, C5×D5, C5×D5, C5⋊D5, C5×C10, C5×C10, D4×D5, C2×C5⋊D4, C5×Dic5, C526C4, D52, D5×C10, D5×C10, C2×C5⋊D5, C102, D5×Dic5, C522D4, C5⋊D20, C5×C5⋊D4, C527D4, C2×D52, D5×C2×C10, D5×C5⋊D4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, D10, C5⋊D4, C22×D5, D4×D5, C2×C5⋊D4, D52, C2×D52, D5×C5⋊D4

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

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

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

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

52 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 5A 5B 5C 5D 5E 5F 5G 5H 10A ··· 10H 10I ··· 10V 10W ··· 10AD 10AE 10AF 20A 20B order 1 2 2 2 2 2 2 2 4 4 5 5 5 5 5 5 5 5 10 ··· 10 10 ··· 10 10 ··· 10 10 10 20 20 size 1 1 2 5 5 10 10 50 10 50 2 2 2 2 4 4 4 4 2 ··· 2 4 ··· 4 10 ··· 10 20 20 20 20

52 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 D4 D5 D5 D10 D10 D10 C5⋊D4 D4×D5 D52 C2×D52 D5×C5⋊D4 kernel D5×C5⋊D4 D5×Dic5 C52⋊2D4 C5⋊D20 C5×C5⋊D4 C52⋊7D4 C2×D52 D5×C2×C10 C5×D5 C5⋊D4 C22×D5 Dic5 D10 C2×C10 D5 C5 C22 C2 C1 # reps 1 1 1 1 1 1 1 1 2 2 2 2 6 4 8 2 4 4 8

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

 1 0 0 0 0 1 0 0 0 0 0 40 0 0 1 34
,
 1 0 0 0 0 1 0 0 0 0 1 34 0 0 0 40
,
 16 0 0 0 0 18 0 0 0 0 1 0 0 0 0 1
,
 0 25 0 0 18 0 0 0 0 0 1 0 0 0 0 1
,
 0 25 0 0 23 0 0 0 0 0 1 0 0 0 0 1
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,0,1,0,0,40,34],[1,0,0,0,0,1,0,0,0,0,1,0,0,0,34,40],[16,0,0,0,0,18,0,0,0,0,1,0,0,0,0,1],[0,18,0,0,25,0,0,0,0,0,1,0,0,0,0,1],[0,23,0,0,25,0,0,0,0,0,1,0,0,0,0,1] >;

D5×C5⋊D4 in GAP, Magma, Sage, TeX

D_5\times C_5\rtimes D_4
% in TeX

G:=Group("D5xC5:D4");
// GroupNames label

G:=SmallGroup(400,179);
// by ID

G=gap.SmallGroup(400,179);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-5,-5,116,970,11525]);
// Polycyclic

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

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