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## G = C2×C5⋊D5order 100 = 22·52

### Direct product of C2 and C5⋊D5

Aliases: C2×C5⋊D5, C10⋊D5, C52D10, C523C22, (C5×C10)⋊2C2, SmallGroup(100,15)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — C2×C5⋊D5
 Chief series C1 — C5 — C52 — C5⋊D5 — C2×C5⋊D5
 Lower central C52 — C2×C5⋊D5
 Upper central C1 — C2

Generators and relations for C2×C5⋊D5
G = < a,b,c,d | a2=b5=c5=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >

Character table of C2×C5⋊D5

 class 1 2A 2B 2C 5A 5B 5C 5D 5E 5F 5G 5H 5I 5J 5K 5L 10A 10B 10C 10D 10E 10F 10G 10H 10I 10J 10K 10L size 1 1 25 25 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ3 1 1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 2 ρ4 1 -1 1 -1 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ5 2 -2 0 0 2 -1+√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1+√5/2 2 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 1-√5/2 -2 1-√5/2 1+√5/2 1+√5/2 1+√5/2 1-√5/2 -2 1-√5/2 1+√5/2 1-√5/2 1+√5/2 orthogonal lifted from D10 ρ6 2 2 0 0 -1-√5/2 -1+√5/2 2 -1+√5/2 -1+√5/2 2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1+√5/2 2 -1+√5/2 -1+√5/2 2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1-√5/2 orthogonal lifted from D5 ρ7 2 -2 0 0 2 -1-√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1-√5/2 2 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 1+√5/2 -2 1+√5/2 1-√5/2 1-√5/2 1-√5/2 1+√5/2 -2 1+√5/2 1-√5/2 1+√5/2 1-√5/2 orthogonal lifted from D10 ρ8 2 -2 0 0 -1+√5/2 2 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 2 -1+√5/2 -1-√5/2 -1-√5/2 1+√5/2 1-√5/2 -2 1-√5/2 1+√5/2 1-√5/2 1+√5/2 1+√5/2 1-√5/2 -2 1-√5/2 1+√5/2 orthogonal lifted from D10 ρ9 2 2 0 0 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 2 2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 orthogonal lifted from D5 ρ10 2 -2 0 0 -1-√5/2 2 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 2 -1-√5/2 -1+√5/2 -1+√5/2 1-√5/2 1+√5/2 -2 1+√5/2 1-√5/2 1+√5/2 1-√5/2 1-√5/2 1+√5/2 -2 1+√5/2 1-√5/2 orthogonal lifted from D10 ρ11 2 -2 0 0 -1-√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1+√5/2 2 2 -1-√5/2 1+√5/2 1+√5/2 1+√5/2 1+√5/2 1+√5/2 1-√5/2 1-√5/2 1-√5/2 1-√5/2 1-√5/2 -2 -2 orthogonal lifted from D10 ρ12 2 -2 0 0 -1-√5/2 -1+√5/2 2 -1+√5/2 -1+√5/2 2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1-√5/2 1+√5/2 1+√5/2 1-√5/2 -2 1-√5/2 1-√5/2 -2 1-√5/2 1+√5/2 1+√5/2 1-√5/2 1+√5/2 orthogonal lifted from D10 ρ13 2 -2 0 0 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 2 -2 1+√5/2 1-√5/2 1-√5/2 1+√5/2 -2 1+√5/2 1-√5/2 1-√5/2 1+√5/2 1+√5/2 1-√5/2 orthogonal lifted from D10 ρ14 2 2 0 0 -1-√5/2 -1-√5/2 -1+√5/2 2 -1-√5/2 -1-√5/2 -1+√5/2 2 -1+√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 2 -1-√5/2 -1-√5/2 -1+√5/2 2 -1+√5/2 -1+√5/2 -1-√5/2 orthogonal lifted from D5 ρ15 2 2 0 0 -1-√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1+√5/2 2 2 -1-√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1+√5/2 2 2 orthogonal lifted from D5 ρ16 2 2 0 0 2 -1+√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1+√5/2 2 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1+√5/2 2 -1+√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1+√5/2 2 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 orthogonal lifted from D5 ρ17 2 -2 0 0 -1-√5/2 -1-√5/2 -1+√5/2 2 -1-√5/2 -1-√5/2 -1+√5/2 2 -1+√5/2 -1+√5/2 -1-√5/2 -1+√5/2 1-√5/2 1+√5/2 1+√5/2 1-√5/2 -2 1+√5/2 1+√5/2 1-√5/2 -2 1-√5/2 1-√5/2 1+√5/2 orthogonal lifted from D10 ρ18 2 2 0 0 -1+√5/2 -1+√5/2 -1-√5/2 2 -1+√5/2 -1+√5/2 -1-√5/2 2 -1-√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 2 -1+√5/2 -1+√5/2 -1-√5/2 2 -1-√5/2 -1-√5/2 -1+√5/2 orthogonal lifted from D5 ρ19 2 -2 0 0 -1+√5/2 -1+√5/2 -1-√5/2 2 -1+√5/2 -1+√5/2 -1-√5/2 2 -1-√5/2 -1-√5/2 -1+√5/2 -1-√5/2 1+√5/2 1-√5/2 1-√5/2 1+√5/2 -2 1-√5/2 1-√5/2 1+√5/2 -2 1+√5/2 1+√5/2 1-√5/2 orthogonal lifted from D10 ρ20 2 2 0 0 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 2 2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 orthogonal lifted from D5 ρ21 2 2 0 0 -1+√5/2 -1-√5/2 2 -1-√5/2 -1-√5/2 2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1-√5/2 2 -1-√5/2 -1-√5/2 2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1+√5/2 orthogonal lifted from D5 ρ22 2 -2 0 0 -1+√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1-√5/2 2 2 -1+√5/2 1-√5/2 1-√5/2 1-√5/2 1-√5/2 1-√5/2 1+√5/2 1+√5/2 1+√5/2 1+√5/2 1+√5/2 -2 -2 orthogonal lifted from D10 ρ23 2 2 0 0 -1+√5/2 2 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 2 -1+√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1+√5/2 2 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 2 -1+√5/2 -1-√5/2 orthogonal lifted from D5 ρ24 2 2 0 0 -1-√5/2 2 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 2 -1-√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1-√5/2 2 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 2 -1-√5/2 -1+√5/2 orthogonal lifted from D5 ρ25 2 -2 0 0 -1+√5/2 -1-√5/2 2 -1-√5/2 -1-√5/2 2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1+√5/2 1-√5/2 1-√5/2 1+√5/2 -2 1+√5/2 1+√5/2 -2 1+√5/2 1-√5/2 1-√5/2 1+√5/2 1-√5/2 orthogonal lifted from D10 ρ26 2 2 0 0 2 -1-√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1-√5/2 2 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1-√5/2 2 -1-√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1-√5/2 2 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 orthogonal lifted from D5 ρ27 2 -2 0 0 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 2 -2 1-√5/2 1+√5/2 1+√5/2 1-√5/2 -2 1-√5/2 1+√5/2 1+√5/2 1-√5/2 1-√5/2 1+√5/2 orthogonal lifted from D10 ρ28 2 2 0 0 -1+√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1-√5/2 2 2 -1+√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1-√5/2 2 2 orthogonal lifted from D5

Smallest permutation representation of C2×C5⋊D5
On 50 points
Generators in S50
(1 29)(2 30)(3 26)(4 27)(5 28)(6 31)(7 32)(8 33)(9 34)(10 35)(11 36)(12 37)(13 38)(14 39)(15 40)(16 41)(17 42)(18 43)(19 44)(20 45)(21 46)(22 47)(23 48)(24 49)(25 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 24 19 14 9)(2 25 20 15 10)(3 21 16 11 6)(4 22 17 12 7)(5 23 18 13 8)(26 46 41 36 31)(27 47 42 37 32)(28 48 43 38 33)(29 49 44 39 34)(30 50 45 40 35)
(1 9)(2 8)(3 7)(4 6)(5 10)(11 22)(12 21)(13 25)(14 24)(15 23)(16 17)(18 20)(26 32)(27 31)(28 35)(29 34)(30 33)(36 47)(37 46)(38 50)(39 49)(40 48)(41 42)(43 45)

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

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

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

C2×C5⋊D5 is a maximal subgroup of   Dic52D5  C5⋊D20  C20⋊D5  C527D4  C2×D52
C2×C5⋊D5 is a maximal quotient of   C524Q8  C20⋊D5  C527D4

Matrix representation of C2×C5⋊D5 in GL4(𝔽11) generated by

 10 0 0 0 0 10 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 0 1 0 0 10 7
,
 0 1 0 0 10 3 0 0 0 0 1 0 0 0 0 1
,
 0 1 0 0 1 0 0 0 0 0 10 7 0 0 0 1
G:=sub<GL(4,GF(11))| [10,0,0,0,0,10,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,0,10,0,0,1,7],[0,10,0,0,1,3,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,10,0,0,0,7,1] >;

C2×C5⋊D5 in GAP, Magma, Sage, TeX

C_2\times C_5\rtimes D_5
% in TeX

G:=Group("C2xC5:D5");
// GroupNames label

G:=SmallGroup(100,15);
// by ID

G=gap.SmallGroup(100,15);
# by ID

G:=PCGroup([4,-2,-2,-5,-5,194,1283]);
// Polycyclic

G:=Group<a,b,c,d|a^2=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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