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## G = C2×C5⋊F5order 200 = 23·52

### Direct product of C2 and C5⋊F5

Aliases: C2×C5⋊F5, C101F5, C5⋊D53C4, C52(C2×F5), (C5×C10)⋊3C4, C525(C2×C4), C5⋊D5.4C22, (C2×C5⋊D5).2C2, SmallGroup(200,47)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — C2×C5⋊F5
 Chief series C1 — C5 — C52 — C5⋊D5 — C5⋊F5 — C2×C5⋊F5
 Lower central C52 — C2×C5⋊F5
 Upper central C1 — C2

Generators and relations for C2×C5⋊F5
G = < a,b,c,d | a2=b5=c5=d4=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b3, dcd-1=c3 >

Subgroups: 352 in 64 conjugacy classes, 22 normal (8 characteristic)
C1, C2, C2 [×2], C4 [×2], C22, C5 [×6], C2×C4, D5 [×12], C10 [×6], F5 [×12], D10 [×6], C52, C2×F5 [×6], C5⋊D5 [×2], C5×C10, C5⋊F5 [×2], C2×C5⋊D5, C2×C5⋊F5
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, F5 [×6], C2×F5 [×6], C5⋊F5, C2×C5⋊F5

Character table of C2×C5⋊F5

 class 1 2A 2B 2C 4A 4B 4C 4D 5A 5B 5C 5D 5E 5F 10A 10B 10C 10D 10E 10F size 1 1 25 25 25 25 25 25 4 4 4 4 4 4 4 4 4 4 4 4 ρ1 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 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 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 linear of order 2 ρ5 1 -1 -1 1 i -i -i i 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 linear of order 4 ρ6 1 1 -1 -1 i i -i -i 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 4 ρ7 1 -1 -1 1 -i i i -i 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 linear of order 4 ρ8 1 1 -1 -1 -i -i i i 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 4 ρ9 4 -4 0 0 0 0 0 0 -1 -1 -1 -1 4 -1 -4 1 1 1 1 1 orthogonal lifted from C2×F5 ρ10 4 -4 0 0 0 0 0 0 -1 -1 -1 4 -1 -1 1 1 1 1 1 -4 orthogonal lifted from C2×F5 ρ11 4 4 0 0 0 0 0 0 4 -1 -1 -1 -1 -1 -1 -1 4 -1 -1 -1 orthogonal lifted from F5 ρ12 4 -4 0 0 0 0 0 0 4 -1 -1 -1 -1 -1 1 1 -4 1 1 1 orthogonal lifted from C2×F5 ρ13 4 4 0 0 0 0 0 0 -1 -1 -1 -1 4 -1 4 -1 -1 -1 -1 -1 orthogonal lifted from F5 ρ14 4 4 0 0 0 0 0 0 -1 -1 -1 4 -1 -1 -1 -1 -1 -1 -1 4 orthogonal lifted from F5 ρ15 4 -4 0 0 0 0 0 0 -1 -1 -1 -1 -1 4 1 -4 1 1 1 1 orthogonal lifted from C2×F5 ρ16 4 -4 0 0 0 0 0 0 -1 4 -1 -1 -1 -1 1 1 1 -4 1 1 orthogonal lifted from C2×F5 ρ17 4 -4 0 0 0 0 0 0 -1 -1 4 -1 -1 -1 1 1 1 1 -4 1 orthogonal lifted from C2×F5 ρ18 4 4 0 0 0 0 0 0 -1 -1 4 -1 -1 -1 -1 -1 -1 -1 4 -1 orthogonal lifted from F5 ρ19 4 4 0 0 0 0 0 0 -1 -1 -1 -1 -1 4 -1 4 -1 -1 -1 -1 orthogonal lifted from F5 ρ20 4 4 0 0 0 0 0 0 -1 4 -1 -1 -1 -1 -1 -1 -1 4 -1 -1 orthogonal lifted from F5

Smallest permutation representation of C2×C5⋊F5
On 50 points
Generators in S50
(1 12)(2 13)(3 14)(4 15)(5 11)(6 30)(7 26)(8 27)(9 28)(10 29)(16 40)(17 36)(18 37)(19 38)(20 39)(21 45)(22 41)(23 42)(24 43)(25 44)(31 48)(32 49)(33 50)(34 46)(35 47)
(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 42 28 39 33)(2 43 29 40 34)(3 44 30 36 35)(4 45 26 37 31)(5 41 27 38 32)(6 17 47 14 25)(7 18 48 15 21)(8 19 49 11 22)(9 20 50 12 23)(10 16 46 13 24)
(2 3 5 4)(6 49 18 24)(7 46 17 22)(8 48 16 25)(9 50 20 23)(10 47 19 21)(11 15 13 14)(26 34 36 41)(27 31 40 44)(28 33 39 42)(29 35 38 45)(30 32 37 43)

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

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

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

C2×C5⋊F5 is a maximal subgroup of   C523C42  D10⋊F5  Dic5⋊F5  C20⋊F5  C102⋊C4
C2×C5⋊F5 is a maximal quotient of   C20.F5  C527M4(2)  C20⋊F5  C5213M4(2)  C102⋊C4

Matrix representation of C2×C5⋊F5 in GL9(𝔽41)

 40 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 40 40 40 40 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 40 40 40 40 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0
,
 1 0 0 0 0 0 0 0 0 0 40 40 40 40 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 40 40 40 40 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0
,
 9 0 0 0 0 0 0 0 0 0 2 0 1 1 0 0 0 0 0 1 1 0 2 0 0 0 0 0 40 1 40 0 0 0 0 0 0 39 40 40 39 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 40 40 40 40

G:=sub<GL(9,GF(41))| [40,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,0,0,40,1,0,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,40,0,0,0,0,0,0,0,1,40,0,0,0,0,0,0,0,0,0,0,40,1,0,0,0,0,0,0,0,40,0,1,0,0,0,0,0,0,40,0,0,1,0,0,0,0,0,40,0,0,0],[1,0,0,0,0,0,0,0,0,0,40,1,0,0,0,0,0,0,0,40,0,1,0,0,0,0,0,0,40,0,0,1,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,0,0,40,1,0,0,0,0,0,0,0,40,0,1,0,0,0,0,0,0,40,0,0,1,0,0,0,0,0,40,0,0,0],[9,0,0,0,0,0,0,0,0,0,2,1,40,39,0,0,0,0,0,0,1,1,40,0,0,0,0,0,1,0,40,40,0,0,0,0,0,1,2,0,39,0,0,0,0,0,0,0,0,0,1,0,0,40,0,0,0,0,0,0,0,1,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,40] >;

C2×C5⋊F5 in GAP, Magma, Sage, TeX

C_2\times C_5\rtimes F_5
% in TeX

G:=Group("C2xC5:F5");
// GroupNames label

G:=SmallGroup(200,47);
// by ID

G=gap.SmallGroup(200,47);
# by ID

G:=PCGroup([5,-2,-2,-2,-5,-5,20,323,173,2004,1014]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^5=c^5=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^3,d*c*d^-1=c^3>;
// generators/relations

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