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## G = C5×C5⋊D5order 250 = 2·53

### Direct product of C5 and C5⋊D5

Aliases: C5×C5⋊D5, C532C2, C523D5, C524C10, C5⋊(C5×D5), SmallGroup(250,13)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — C5×C5⋊D5
 Chief series C1 — C5 — C52 — C53 — C5×C5⋊D5
 Lower central C52 — C5×C5⋊D5
 Upper central C1 — C5

Generators and relations for C5×C5⋊D5
G = < a,b,c,d | a5=b5=c5=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >

Subgroups: 176 in 56 conjugacy classes, 18 normal (6 characteristic)
C1, C2, C5, C5, C5, D5, C10, C52, C52, C52, C5×D5, C5⋊D5, C53, C5×C5⋊D5
Quotients: C1, C2, C5, D5, C10, C5×D5, C5⋊D5, C5×C5⋊D5

Smallest permutation representation of C5×C5⋊D5
On 50 points
Generators in S50
(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)(41 42 43 44 45)(46 47 48 49 50)
(1 39 31 30 45)(2 40 32 26 41)(3 36 33 27 42)(4 37 34 28 43)(5 38 35 29 44)(6 12 50 20 21)(7 13 46 16 22)(8 14 47 17 23)(9 15 48 18 24)(10 11 49 19 25)
(1 36 35 26 43)(2 37 31 27 44)(3 38 32 28 45)(4 39 33 29 41)(5 40 34 30 42)(6 15 46 19 23)(7 11 47 20 24)(8 12 48 16 25)(9 13 49 17 21)(10 14 50 18 22)
(1 47)(2 48)(3 49)(4 50)(5 46)(6 34)(7 35)(8 31)(9 32)(10 33)(11 36)(12 37)(13 38)(14 39)(15 40)(16 44)(17 45)(18 41)(19 42)(20 43)(21 28)(22 29)(23 30)(24 26)(25 27)

G:=sub<Sym(50)| (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)(41,42,43,44,45)(46,47,48,49,50), (1,39,31,30,45)(2,40,32,26,41)(3,36,33,27,42)(4,37,34,28,43)(5,38,35,29,44)(6,12,50,20,21)(7,13,46,16,22)(8,14,47,17,23)(9,15,48,18,24)(10,11,49,19,25), (1,36,35,26,43)(2,37,31,27,44)(3,38,32,28,45)(4,39,33,29,41)(5,40,34,30,42)(6,15,46,19,23)(7,11,47,20,24)(8,12,48,16,25)(9,13,49,17,21)(10,14,50,18,22), (1,47)(2,48)(3,49)(4,50)(5,46)(6,34)(7,35)(8,31)(9,32)(10,33)(11,36)(12,37)(13,38)(14,39)(15,40)(16,44)(17,45)(18,41)(19,42)(20,43)(21,28)(22,29)(23,30)(24,26)(25,27)>;

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)(41,42,43,44,45)(46,47,48,49,50), (1,39,31,30,45)(2,40,32,26,41)(3,36,33,27,42)(4,37,34,28,43)(5,38,35,29,44)(6,12,50,20,21)(7,13,46,16,22)(8,14,47,17,23)(9,15,48,18,24)(10,11,49,19,25), (1,36,35,26,43)(2,37,31,27,44)(3,38,32,28,45)(4,39,33,29,41)(5,40,34,30,42)(6,15,46,19,23)(7,11,47,20,24)(8,12,48,16,25)(9,13,49,17,21)(10,14,50,18,22), (1,47)(2,48)(3,49)(4,50)(5,46)(6,34)(7,35)(8,31)(9,32)(10,33)(11,36)(12,37)(13,38)(14,39)(15,40)(16,44)(17,45)(18,41)(19,42)(20,43)(21,28)(22,29)(23,30)(24,26)(25,27) );

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),(41,42,43,44,45),(46,47,48,49,50)], [(1,39,31,30,45),(2,40,32,26,41),(3,36,33,27,42),(4,37,34,28,43),(5,38,35,29,44),(6,12,50,20,21),(7,13,46,16,22),(8,14,47,17,23),(9,15,48,18,24),(10,11,49,19,25)], [(1,36,35,26,43),(2,37,31,27,44),(3,38,32,28,45),(4,39,33,29,41),(5,40,34,30,42),(6,15,46,19,23),(7,11,47,20,24),(8,12,48,16,25),(9,13,49,17,21),(10,14,50,18,22)], [(1,47),(2,48),(3,49),(4,50),(5,46),(6,34),(7,35),(8,31),(9,32),(10,33),(11,36),(12,37),(13,38),(14,39),(15,40),(16,44),(17,45),(18,41),(19,42),(20,43),(21,28),(22,29),(23,30),(24,26),(25,27)]])

C5×C5⋊D5 is a maximal subgroup of   C536C4  C537C4  C5×D52  C525D10

70 conjugacy classes

 class 1 2 5A 5B 5C 5D 5E ··· 5BL 10A 10B 10C 10D order 1 2 5 5 5 5 5 ··· 5 10 10 10 10 size 1 25 1 1 1 1 2 ··· 2 25 25 25 25

70 irreducible representations

 dim 1 1 1 1 2 2 type + + + image C1 C2 C5 C10 D5 C5×D5 kernel C5×C5⋊D5 C53 C5⋊D5 C52 C52 C5 # reps 1 1 4 4 12 48

Matrix representation of C5×C5⋊D5 in GL4(𝔽11) generated by

 4 0 0 0 0 4 0 0 0 0 1 0 0 0 0 1
,
 5 0 0 0 0 9 0 0 0 0 5 0 0 0 0 9
,
 3 0 0 0 0 4 0 0 0 0 5 0 0 0 0 9
,
 0 4 0 0 3 0 0 0 0 0 0 9 0 0 5 0
G:=sub<GL(4,GF(11))| [4,0,0,0,0,4,0,0,0,0,1,0,0,0,0,1],[5,0,0,0,0,9,0,0,0,0,5,0,0,0,0,9],[3,0,0,0,0,4,0,0,0,0,5,0,0,0,0,9],[0,3,0,0,4,0,0,0,0,0,0,5,0,0,9,0] >;

C5×C5⋊D5 in GAP, Magma, Sage, TeX

C_5\times C_5\rtimes D_5
% in TeX

G:=Group("C5xC5:D5");
// GroupNames label

G:=SmallGroup(250,13);
// by ID

G=gap.SmallGroup(250,13);
# by ID

G:=PCGroup([4,-2,-5,-5,-5,482,3203]);
// Polycyclic

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

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