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G = C2×C5⋊D5  order 100 = 22·52

Direct product of C2 and C5⋊D5

direct product, metabelian, supersoluble, monomial, A-group

Aliases: C2×C5⋊D5, C10⋊D5, C5⋊2D10, C52⋊3C22, (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 >

C1
C2
25C2
25C2
C5
C5
C5
C5
C5
C5
25C22
C10
C10
C10
C10
C10
C10
5D5
5D5
5D5
5D5
5D5
5D5
5D5
5D5
5D5
5D5
5D5
5D5
C52
5D10
5D10
5D10
5D10
5D10
5D10
C5⋊D5
C5⋊D5
C5×C10
C2×C5⋊D5

Character table of C2×C5⋊D5

 class 12A2B2C5A5B5C5D5E5F5G5H5I5J5K5L10A10B10C10D10E10F10G10H10I10J10K10L
 size 112525222222222222222222222222
ρ11111111111111111111111111111    trivial
ρ21-1-11111111111111-1-1-1-1-1-1-1-1-1-1-1-1    linear of order 2
ρ311-1-1111111111111111111111111    linear of order 2
ρ41-11-1111111111111-1-1-1-1-1-1-1-1-1-1-1-1    linear of order 2
ρ52-2002-1+√5/2-1-√5/2-1-√5/2-1-√5/2-1+√5/22-1+√5/2-1-√5/2-1+√5/2-1-√5/2-1+√5/21-√5/2-21-√5/21+√5/21+√5/21+√5/21-√5/2-21-√5/21+√5/21-√5/21+√5/2    orthogonal lifted from D10
ρ62200-1-√5/2-1+√5/22-1+√5/2-1+√5/22-1+√5/2-1-√5/2-1-√5/2-1+√5/2-1-√5/2-1-√5/2-1-√5/2-1-√5/2-1+√5/22-1+√5/2-1+√5/22-1+√5/2-1-√5/2-1-√5/2-1+√5/2-1-√5/2    orthogonal lifted from D5
ρ72-2002-1-√5/2-1+√5/2-1+√5/2-1+√5/2-1-√5/22-1-√5/2-1+√5/2-1-√5/2-1+√5/2-1-√5/21+√5/2-21+√5/21-√5/21-√5/21-√5/21+√5/2-21+√5/21-√5/21+√5/21-√5/2    orthogonal lifted from D10
ρ82-200-1+√5/22-1+√5/2-1-√5/2-1+√5/2-1-√5/2-1-√5/2-1+√5/22-1+√5/2-1-√5/2-1-√5/21+√5/21-√5/2-21-√5/21+√5/21-√5/21+√5/21+√5/21-√5/2-21-√5/21+√5/2    orthogonal lifted from D10
ρ92200-1+√5/2-1-√5/2-1-√5/2-1+√5/22-1+√5/2-1-√5/2-1-√5/2-1+√5/2-1+√5/2-1-√5/222-1+√5/2-1-√5/2-1-√5/2-1+√5/22-1+√5/2-1-√5/2-1-√5/2-1+√5/2-1+√5/2-1-√5/2    orthogonal lifted from D5
ρ102-200-1-√5/22-1-√5/2-1+√5/2-1-√5/2-1+√5/2-1+√5/2-1-√5/22-1-√5/2-1+√5/2-1+√5/21-√5/21+√5/2-21+√5/21-√5/21+√5/21-√5/21-√5/21+√5/2-21+√5/21-√5/2    orthogonal lifted from D10
ρ112-200-1-√5/2-1-√5/2-1-√5/2-1-√5/2-1+√5/2-1+√5/2-1+√5/2-1+√5/2-1+√5/222-1-√5/21+√5/21+√5/21+√5/21+√5/21+√5/21-√5/21-√5/21-√5/21-√5/21-√5/2-2-2    orthogonal lifted from D10
ρ122-200-1-√5/2-1+√5/22-1+√5/2-1+√5/22-1+√5/2-1-√5/2-1-√5/2-1+√5/2-1-√5/2-1-√5/21+√5/21+√5/21-√5/2-21-√5/21-√5/2-21-√5/21+√5/21+√5/21-√5/21+√5/2    orthogonal lifted from D10
ρ132-200-1-√5/2-1+√5/2-1+√5/2-1-√5/22-1-√5/2-1+√5/2-1+√5/2-1-√5/2-1-√5/2-1+√5/22-21+√5/21-√5/21-√5/21+√5/2-21+√5/21-√5/21-√5/21+√5/21+√5/21-√5/2    orthogonal lifted from D10
ρ142200-1-√5/2-1-√5/2-1+√5/22-1-√5/2-1-√5/2-1+√5/22-1+√5/2-1+√5/2-1-√5/2-1+√5/2-1+√5/2-1-√5/2-1-√5/2-1+√5/22-1-√5/2-1-√5/2-1+√5/22-1+√5/2-1+√5/2-1-√5/2    orthogonal lifted from D5
ρ152200-1-√5/2-1-√5/2-1-√5/2-1-√5/2-1+√5/2-1+√5/2-1+√5/2-1+√5/2-1+√5/222-1-√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+√5/222    orthogonal lifted from D5
ρ1622002-1+√5/2-1-√5/2-1-√5/2-1-√5/2-1+√5/22-1+√5/2-1-√5/2-1+√5/2-1-√5/2-1+√5/2-1+√5/22-1+√5/2-1-√5/2-1-√5/2-1-√5/2-1+√5/22-1+√5/2-1-√5/2-1+√5/2-1-√5/2    orthogonal lifted from D5
ρ172-200-1-√5/2-1-√5/2-1+√5/22-1-√5/2-1-√5/2-1+√5/22-1+√5/2-1+√5/2-1-√5/2-1+√5/21-√5/21+√5/21+√5/21-√5/2-21+√5/21+√5/21-√5/2-21-√5/21-√5/21+√5/2    orthogonal lifted from D10
ρ182200-1+√5/2-1+√5/2-1-√5/22-1+√5/2-1+√5/2-1-√5/22-1-√5/2-1-√5/2-1+√5/2-1-√5/2-1-√5/2-1+√5/2-1+√5/2-1-√5/22-1+√5/2-1+√5/2-1-√5/22-1-√5/2-1-√5/2-1+√5/2    orthogonal lifted from D5
ρ192-200-1+√5/2-1+√5/2-1-√5/22-1+√5/2-1+√5/2-1-√5/22-1-√5/2-1-√5/2-1+√5/2-1-√5/21+√5/21-√5/21-√5/21+√5/2-21-√5/21-√5/21+√5/2-21+√5/21+√5/21-√5/2    orthogonal lifted from D10
ρ202200-1-√5/2-1+√5/2-1+√5/2-1-√5/22-1-√5/2-1+√5/2-1+√5/2-1-√5/2-1-√5/2-1+√5/222-1-√5/2-1+√5/2-1+√5/2-1-√5/22-1-√5/2-1+√5/2-1+√5/2-1-√5/2-1-√5/2-1+√5/2    orthogonal lifted from D5
ρ212200-1+√5/2-1-√5/22-1-√5/2-1-√5/22-1-√5/2-1+√5/2-1+√5/2-1-√5/2-1+√5/2-1+√5/2-1+√5/2-1+√5/2-1-√5/22-1-√5/2-1-√5/22-1-√5/2-1+√5/2-1+√5/2-1-√5/2-1+√5/2    orthogonal lifted from D5
ρ222-200-1+√5/2-1+√5/2-1+√5/2-1+√5/2-1-√5/2-1-√5/2-1-√5/2-1-√5/2-1-√5/222-1+√5/21-√5/21-√5/21-√5/21-√5/21-√5/21+√5/21+√5/21+√5/21+√5/21+√5/2-2-2    orthogonal lifted from D10
ρ232200-1+√5/22-1+√5/2-1-√5/2-1+√5/2-1-√5/2-1-√5/2-1+√5/22-1+√5/2-1-√5/2-1-√5/2-1-√5/2-1+√5/22-1+√5/2-1-√5/2-1+√5/2-1-√5/2-1-√5/2-1+√5/22-1+√5/2-1-√5/2    orthogonal lifted from D5
ρ242200-1-√5/22-1-√5/2-1+√5/2-1-√5/2-1+√5/2-1+√5/2-1-√5/22-1-√5/2-1+√5/2-1+√5/2-1+√5/2-1-√5/22-1-√5/2-1+√5/2-1-√5/2-1+√5/2-1+√5/2-1-√5/22-1-√5/2-1+√5/2    orthogonal lifted from D5
ρ252-200-1+√5/2-1-√5/22-1-√5/2-1-√5/22-1-√5/2-1+√5/2-1+√5/2-1-√5/2-1+√5/2-1+√5/21-√5/21-√5/21+√5/2-21+√5/21+√5/2-21+√5/21-√5/21-√5/21+√5/21-√5/2    orthogonal lifted from D10
ρ2622002-1-√5/2-1+√5/2-1+√5/2-1+√5/2-1-√5/22-1-√5/2-1+√5/2-1-√5/2-1+√5/2-1-√5/2-1-√5/22-1-√5/2-1+√5/2-1+√5/2-1+√5/2-1-√5/22-1-√5/2-1+√5/2-1-√5/2-1+√5/2    orthogonal lifted from D5
ρ272-200-1+√5/2-1-√5/2-1-√5/2-1+√5/22-1+√5/2-1-√5/2-1-√5/2-1+√5/2-1+√5/2-1-√5/22-21-√5/21+√5/21+√5/21-√5/2-21-√5/21+√5/21+√5/21-√5/21-√5/21+√5/2    orthogonal lifted from D10
ρ282200-1+√5/2-1+√5/2-1+√5/2-1+√5/2-1-√5/2-1-√5/2-1-√5/2-1-√5/2-1-√5/222-1+√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-√5/222    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   Dic5⋊2D5  C5⋊D20  C20⋊D5  C52⋊7D4  C2×D52
C2×C5⋊D5 is a maximal quotient of   C52⋊4Q8  C20⋊D5  C52⋊7D4

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

10000
01000
0010
0001
,
1000
0100
0001
00107
,
0100
10300
0010
0001
,
0100
1000
00107
0001
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
 

Export

Subgroup lattice of C2×C5⋊D5 in TeX
Character table of C2×C5⋊D5 in TeX

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