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G = F52order 400 = 24·52

Direct product of F5 and F5

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

Aliases: F52, C52⋊C42, (C5×F5)⋊C4, D5.D5⋊C4, C52⋊C4⋊C4, C5⋊F5⋊C4, C54(C4×F5), (D5×F5).C2, D5.(C2×F5), D5⋊F5.1C2, D52.1C22, (C5×D5).(C2×C4), C5⋊D5.1(C2×C4), Hol(F5), SmallGroup(400,205)

Series: Derived Chief Lower central Upper central

C1C52 — F52
C1C5C52C5×D5D52D5×F5 — F52
C52 — F52
C1

Generators and relations for F52
 G = < a,b,c,d | a5=b4=c5=d4=1, bab-1=a3, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c3 >

Subgroups: 456 in 63 conjugacy classes, 22 normal (8 characteristic)
C1, C2 [×3], C4 [×6], C22, C5 [×2], C5, C2×C4 [×3], D5 [×2], D5 [×3], C10 [×2], C42, Dic5 [×2], C20 [×2], F5 [×2], F5 [×7], D10 [×2], C52, C4×D5 [×2], C2×F5 [×4], C5×D5 [×2], C5⋊D5, C4×F5 [×2], C5×F5 [×2], D5.D5 [×2], C5⋊F5, C52⋊C4, D52, D5×F5 [×2], D5⋊F5, F52
Quotients: C1, C2 [×3], C4 [×6], C22, C2×C4 [×3], C42, F5 [×2], C2×F5 [×2], C4×F5 [×2], F52

Character table of F52

 class 12A2B2C4A4B4C4D4E4F4G4H4I4J4K4L5A5B5C10A10B20A20B20C20D
 size 15525555525252525252525254416202020202020
ρ11111111111111111111111111    trivial
ρ211111-11-1-11-11-1-1-1-111111-1-111    linear of order 2
ρ31111-1-1-1-11-1-1-1111-111111-1-1-1-1    linear of order 2
ρ41111-11-11-1-11-1-1-1-111111111-1-1    linear of order 2
ρ51-11-1-1-i-1ii1-i1-ii-ii111-11-ii-1-1    linear of order 4
ρ61-11-1-1i-1-i-i1i1i-ii-i111-11i-i-1-1    linear of order 4
ρ71-1-11-i-iii-1-iii-111-i111-1-1-ii-ii    linear of order 4
ρ811-1-1i-1-i-1-i-i1iii-i11111-1-1-1i-i    linear of order 4
ρ911-1-1i1-i1i-i-1i-i-ii-11111-111i-i    linear of order 4
ρ101-1-11-iii-i1-i-ii1-1-1i111-1-1i-i-ii    linear of order 4
ρ111-11-11i1-ii-1i-1-ii-i-i111-11i-i11    linear of order 4
ρ121-11-11-i1i-i-1-i-1i-iii111-11-ii11    linear of order 4
ρ1311-1-1-i1i1-ii-1-iii-i-11111-111-ii    linear of order 4
ρ141-1-11ii-i-i-1i-i-i-111i111-1-1i-ii-i    linear of order 4
ρ1511-1-1-i-1i-1ii1-i-i-ii11111-1-1-1-ii    linear of order 4
ρ161-1-11i-i-ii1ii-i1-1-1-i111-1-1-iii-i    linear of order 4
ρ174040-40-40000000004-1-10-10011    orthogonal lifted from C2×F5
ρ1844000-40-400000000-14-1-101100    orthogonal lifted from C2×F5
ρ1940404040000000004-1-10-100-1-1    orthogonal lifted from F5
ρ204400040400000000-14-1-10-1-100    orthogonal lifted from F5
ρ2140-40-4i04i0000000004-1-10100i-i    complex lifted from C4×F5
ρ224-4000-4i04i00000000-14-110i-i00    complex lifted from C4×F5
ρ2340-404i0-4i0000000004-1-10100-ii    complex lifted from C4×F5
ρ244-40004i0-4i00000000-14-110-ii00    complex lifted from C4×F5
ρ2516000000000000000-4-41000000    orthogonal faithful

Permutation representations of F52
On 20 points - transitive group 20T102
Generators in S20
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)
(1 18 6 12)(2 20 10 15)(3 17 9 13)(4 19 8 11)(5 16 7 14)
(1 5 4 3 2)(6 7 8 9 10)(11 13 15 12 14)(16 19 17 20 18)
(1 12 6 18)(2 13 7 19)(3 14 8 20)(4 15 9 16)(5 11 10 17)

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

G:=Group( (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,18,6,12)(2,20,10,15)(3,17,9,13)(4,19,8,11)(5,16,7,14), (1,5,4,3,2)(6,7,8,9,10)(11,13,15,12,14)(16,19,17,20,18), (1,12,6,18)(2,13,7,19)(3,14,8,20)(4,15,9,16)(5,11,10,17) );

G=PermutationGroup([(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,17,18,19,20)], [(1,18,6,12),(2,20,10,15),(3,17,9,13),(4,19,8,11),(5,16,7,14)], [(1,5,4,3,2),(6,7,8,9,10),(11,13,15,12,14),(16,19,17,20,18)], [(1,12,6,18),(2,13,7,19),(3,14,8,20),(4,15,9,16),(5,11,10,17)])

G:=TransitiveGroup(20,102);

On 25 points - transitive group 25T32
Generators in S25
(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)
(2 3 5 4)(6 10 8 9)(11 13 12 15)(16 19 20 17)(21 24 25 22)
(1 18 23 14 7)(2 19 24 15 8)(3 20 25 11 9)(4 16 21 12 10)(5 17 22 13 6)
(6 13 17 22)(7 14 18 23)(8 15 19 24)(9 11 20 25)(10 12 16 21)

G:=sub<Sym(25)| (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), (2,3,5,4)(6,10,8,9)(11,13,12,15)(16,19,20,17)(21,24,25,22), (1,18,23,14,7)(2,19,24,15,8)(3,20,25,11,9)(4,16,21,12,10)(5,17,22,13,6), (6,13,17,22)(7,14,18,23)(8,15,19,24)(9,11,20,25)(10,12,16,21)>;

G:=Group( (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), (2,3,5,4)(6,10,8,9)(11,13,12,15)(16,19,20,17)(21,24,25,22), (1,18,23,14,7)(2,19,24,15,8)(3,20,25,11,9)(4,16,21,12,10)(5,17,22,13,6), (6,13,17,22)(7,14,18,23)(8,15,19,24)(9,11,20,25)(10,12,16,21) );

G=PermutationGroup([(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)], [(2,3,5,4),(6,10,8,9),(11,13,12,15),(16,19,20,17),(21,24,25,22)], [(1,18,23,14,7),(2,19,24,15,8),(3,20,25,11,9),(4,16,21,12,10),(5,17,22,13,6)], [(6,13,17,22),(7,14,18,23),(8,15,19,24),(9,11,20,25),(10,12,16,21)])

G:=TransitiveGroup(25,32);

Matrix representation of F52 in GL8(𝔽41)

401000000
400010000
400000000
400100000
00001000
00000100
00000010
00000001
,
000400000
400000000
040000000
004000000
000032000
000003200
000000320
000000032
,
10000000
01000000
00100000
00010000
00000100
00000001
000040404040
00000010
,
400000000
040000000
004000000
000400000
00000009
00009000
00000900
00000090

G:=sub<GL(8,GF(41))| [40,40,40,40,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,32],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,1,0,40,0,0,0,0,0,0,0,40,1,0,0,0,0,0,1,40,0],[40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,9,0,0,0,0,9,0,0,0] >;

F52 in GAP, Magma, Sage, TeX

F_5^2
% in TeX

G:=Group("F5^2");
// GroupNames label

G:=SmallGroup(400,205);
// by ID

G=gap.SmallGroup(400,205);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-5,-5,24,55,490,262,5765,2897]);
// Polycyclic

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

Export

Character table of F52 in TeX

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