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G = C2×C5⋊F5  order 200 = 23·52

Direct product of C2 and C5⋊F5

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

Aliases: C2×C5⋊F5, C10⋊1F5, C5⋊D5⋊3C4, C5⋊2(C2×F5), (C5×C10)⋊3C4, C52⋊5(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, C4, C22, C5, C2×C4, D5, C10, F5, D10, C52, C2×F5, C5⋊D5, C5×C10, C5⋊F5, C2×C5⋊D5, C2×C5⋊F5
Quotients: C1, C2, C4, C22, C2×C4, F5, C2×F5, C5⋊F5, C2×C5⋊F5

Character table of C2×C5⋊F5

 class 12A2B2C4A4B4C4D5A5B5C5D5E5F10A10B10C10D10E10F
 size 11252525252525444444444444
ρ111111111111111111111    trivial
ρ21-11-11-11-1111111-1-1-1-1-1-1    linear of order 2
ρ31111-1-1-1-1111111111111    linear of order 2
ρ41-11-1-11-11111111-1-1-1-1-1-1    linear of order 2
ρ51-1-11i-i-ii111111-1-1-1-1-1-1    linear of order 4
ρ611-1-1ii-i-i111111111111    linear of order 4
ρ71-1-11-iii-i111111-1-1-1-1-1-1    linear of order 4
ρ811-1-1-i-iii111111111111    linear of order 4
ρ94-4000000-1-1-1-14-1-411111    orthogonal lifted from C2×F5
ρ104-4000000-1-1-14-1-111111-4    orthogonal lifted from C2×F5
ρ11440000004-1-1-1-1-1-1-14-1-1-1    orthogonal lifted from F5
ρ124-40000004-1-1-1-1-111-4111    orthogonal lifted from C2×F5
ρ1344000000-1-1-1-14-14-1-1-1-1-1    orthogonal lifted from F5
ρ1444000000-1-1-14-1-1-1-1-1-1-14    orthogonal lifted from F5
ρ154-4000000-1-1-1-1-141-41111    orthogonal lifted from C2×F5
ρ164-4000000-14-1-1-1-1111-411    orthogonal lifted from C2×F5
ρ174-4000000-1-14-1-1-11111-41    orthogonal lifted from C2×F5
ρ1844000000-1-14-1-1-1-1-1-1-14-1    orthogonal lifted from F5
ρ1944000000-1-1-1-1-14-14-1-1-1-1    orthogonal lifted from F5
ρ2044000000-14-1-1-1-1-1-1-14-1-1    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   C52⋊3C42  D10⋊F5  Dic5⋊F5  C20⋊F5  C102⋊C4
C2×C5⋊F5 is a maximal quotient of   C20.F5  C52⋊7M4(2)  C20⋊F5  C52⋊13M4(2)  C102⋊C4

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

4000000000
010000000
001000000
000100000
000010000
000001000
000000100
000000010
000000001
,
100000000
000100000
000010000
0404040400000
010000000
0000040404040
000001000
000000100
000000010
,
100000000
0404040400000
010000000
001000000
000100000
0000040404040
000001000
000000100
000000010
,
900000000
020110000
011020000
04014000000
0394040390000
000001000
000000001
000000100
0000040404040

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
 

Export

Character table of C2×C5⋊F5 in TeX

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