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G = C2×D5⋊F5  order 400 = 24·52

Direct product of C2 and D5⋊F5

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

Aliases: C2×D5⋊F5, D10⋊3F5, D52⋊3C4, D5⋊(C2×F5), C10⋊1(C2×F5), (D5×C10)⋊4C4, D52.4C22, C5⋊F5⋊C22, C5⋊1(C22×F5), C5⋊D5.2C23, C52⋊2(C22×C4), C52⋊C4⋊1C22, C5⋊D5⋊2(C2×C4), (C2×D52).5C2, (C5×C10)⋊2(C2×C4), (C5×D5)⋊2(C2×C4), (C2×C52⋊C4)⋊4C2, (C2×C5⋊F5)⋊3C2, (C2×C5⋊D5).9C22, SmallGroup(400,210)

Series: Derived ►Chief ►Lower central ►Upper central

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

Generators and relations for C2×D5⋊F5
 G = < a,b,c,d,e | a2=b5=c2=d5=e4=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, ebe-1=b3, cd=dc, ece-1=b2c, ede-1=d3 >

Subgroups: 892 in 124 conjugacy classes, 39 normal (11 characteristic)
C1, C2, C2, C4, C22, C5, C5, C2×C4, C23, D5, D5, C10, C10, C22×C4, F5, D10, D10, C2×C10, C52, C2×F5, C22×D5, C5×D5, C5⋊D5, C5×C10, C22×F5, C5⋊F5, C52⋊C4, D52, D5×C10, C2×C5⋊D5, D5⋊F5, C2×C5⋊F5, C2×C52⋊C4, C2×D52, C2×D5⋊F5
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, F5, C2×F5, C22×F5, D5⋊F5, C2×D5⋊F5

Character table of C2×D5⋊F5

 class 12A2B2C2D2E2F2G4A4B4C4D4E4F4G4H5A5B5C5D10A10B10C10D10E10F10G10H
 size 115555252525252525252525254488448820202020
ρ11111111111111111111111111111    trivial
ρ21-1-1-1111-1-1-1-11111-11111-1-1-1-11-1-11    linear of order 2
ρ311111111-1-1-1-1-1-1-1-1111111111111    linear of order 2
ρ41-1-1-1111-1111-1-1-1-111111-1-1-1-11-1-11    linear of order 2
ρ51-111-1-11-1-111-1-111-11111-1-1-1-1-111-1    linear of order 2
ρ611-1-1-1-1111-1-1-1-111111111111-1-1-1-1    linear of order 2
ρ711-1-1-1-111-11111-1-1-111111111-1-1-1-1    linear of order 2
ρ81-111-1-11-11-1-111-1-111111-1-1-1-1-111-1    linear of order 2
ρ9111-1-11-1-1-i-ii-iii-ii11111111-11-11    linear of order 4
ρ101-1-11-11-11ii-i-iii-i-i1111-1-1-1-1-1-111    linear of order 4
ρ111-1-11-11-11-i-iii-i-iii1111-1-1-1-1-1-111    linear of order 4
ρ12111-1-11-1-1ii-ii-i-ii-i11111111-11-11    linear of order 4
ρ1311-111-1-1-1-ii-ii-ii-ii111111111-11-1    linear of order 4
ρ141-11-11-1-11i-iii-ii-i-i1111-1-1-1-111-1-1    linear of order 4
ρ151-11-11-1-11-ii-i-ii-iii1111-1-1-1-111-1-1    linear of order 4
ρ1611-111-1-1-1i-ii-ii-ii-i111111111-11-1    linear of order 4
ρ174-40-4400000000000-14-1-1-4111-1010    orthogonal lifted from C2×F5
ρ184-4-400400000000004-1-1-11-411010-1    orthogonal lifted from C2×F5
ρ19440-4-400000000000-14-1-14-1-1-11010    orthogonal lifted from C2×F5
ρ204-4400-400000000004-1-1-11-4110-101    orthogonal lifted from C2×F5
ρ214404400000000000-14-1-14-1-1-1-10-10    orthogonal lifted from F5
ρ224-404-400000000000-14-1-1-411110-10    orthogonal lifted from C2×F5
ρ2344400400000000004-1-1-1-14-1-10-10-1    orthogonal lifted from F5
ρ2444-400-400000000004-1-1-1-14-1-10101    orthogonal lifted from C2×F5
ρ258-800000000000000-2-23-222-320000    orthogonal faithful
ρ268800000000000000-2-23-2-2-23-20000    orthogonal lifted from D5⋊F5
ρ278800000000000000-2-2-23-2-2-230000    orthogonal lifted from D5⋊F5
ρ288-800000000000000-2-2-23222-30000    orthogonal faithful

Permutation representations of C2×D5⋊F5
►On 20 points - transitive group 20T114
Generators in S20
(1 8)(2 9)(3 10)(4 6)(5 7)(11 16)(12 17)(13 18)(14 19)(15 20)

(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)

(1 16)(2 20)(3 19)(4 18)(5 17)(6 13)(7 12)(8 11)(9 15)(10 14)

(1 5 4 3 2)(6 10 9 8 7)(11 12 13 14 15)(16 17 18 19 20)

(2 3 5 4)(6 9 10 7)(11 15 13 14)(16 20 18 19)
 
G:=sub<Sym(20)| (1,8)(2,9)(3,10)(4,6)(5,7)(11,16)(12,17)(13,18)(14,19)(15,20), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,16)(2,20)(3,19)(4,18)(5,17)(6,13)(7,12)(8,11)(9,15)(10,14), (1,5,4,3,2)(6,10,9,8,7)(11,12,13,14,15)(16,17,18,19,20), (2,3,5,4)(6,9,10,7)(11,15,13,14)(16,20,18,19)>;
 
G:=Group( (1,8)(2,9)(3,10)(4,6)(5,7)(11,16)(12,17)(13,18)(14,19)(15,20), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,16)(2,20)(3,19)(4,18)(5,17)(6,13)(7,12)(8,11)(9,15)(10,14), (1,5,4,3,2)(6,10,9,8,7)(11,12,13,14,15)(16,17,18,19,20), (2,3,5,4)(6,9,10,7)(11,15,13,14)(16,20,18,19) );
 
G=PermutationGroup([[(1,8),(2,9),(3,10),(4,6),(5,7),(11,16),(12,17),(13,18),(14,19),(15,20)], [(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,17,18,19,20)], [(1,16),(2,20),(3,19),(4,18),(5,17),(6,13),(7,12),(8,11),(9,15),(10,14)], [(1,5,4,3,2),(6,10,9,8,7),(11,12,13,14,15),(16,17,18,19,20)], [(2,3,5,4),(6,9,10,7),(11,15,13,14),(16,20,18,19)]])
 
G:=TransitiveGroup(20,114);
 

Matrix representation of C2×D5⋊F5 ►in GL8(ℤ)

-10000000
0-1000000
00-100000
000-10000
0000-1000
00000-100
000000-10
0000000-1
,
00010000
-1-1-1-10000
10000000
01000000
00000010
00000001
0000-1-1-1-1
00001000
,
000000-10
0000000-1
00001111
0000-1000
000-10000
11110000
-10000000
0-1000000
,
00100000
00010000
-1-1-1-10000
10000000
00000010
00000001
0000-1-1-1-1
00001000
,
-10000000
000-10000
0-1000000
11110000
0000-1000
0000000-1
00000-100
00001111

G:=sub<GL(8,Integers())| [-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1],[0,-1,1,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,-1,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,1,-1,0],[0,0,0,0,0,1,-1,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,1,0,0,0,0,0,0,-1,1,0,0,0,0,1,-1,0,0,0,0,0,0,1,0,0,0,0,0,-1,0,1,0,0,0,0,0,0,-1,1,0,0,0,0,0],[0,0,-1,1,0,0,0,0,0,0,-1,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,-1,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,1,-1,0],[-1,0,0,1,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,1,0,0,0,0,0,-1,0,1,0,0,0,0,0,0,0,0,-1,0,0,1,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,1,0,0,0,0,0,-1,0,1] >;
 

C2×D5⋊F5 in GAP, Magma, Sage, TeX 

C_2\times D_5\rtimes F_5
 
% in TeX
 
G:=Group("C2xD5:F5");
 
// GroupNames label
 
G:=SmallGroup(400,210);
 
// by ID
 
G=gap.SmallGroup(400,210);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-2,-5,-5,48,1444,970,262,8645,1463]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^5=c^2=d^5=e^4=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,e*b*e^-1=b^3,c*d=d*c,e*c*e^-1=b^2*c,e*d*e^-1=d^3>;
 
// generators/relations
 

Export

Character table of C2×D5⋊F5 in TeX

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