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G = C52⋊3C42  order 400 = 24·52

2nd semidirect product of C52 and C42 acting via C42/C2=C2×C4

metabelian, supersoluble, monomial, A-group

Aliases: Dic5⋊2F5, C52⋊3C42, C5⋊1(C4×F5), C5⋊F5⋊2C4, C52⋊C4⋊2C4, C10.1(C2×F5), (C5×Dic5)⋊4C4, C2.1(D5⋊F5), Dic5⋊2D5.6C2, C5⋊D5.4(C2×C4), (C5×C10).8(C2×C4), (C2×C5⋊F5).1C2, (C2×C52⋊C4).1C2, (C2×C5⋊D5).3C22, SmallGroup(400,124)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C52 — C52⋊3C42
Chief series
C1 — C5 — C52 — C5⋊D5 — C2×C5⋊D5 — C2×C5⋊F5 — C52⋊3C42
Lower central
C52 — C52⋊3C42
Upper central
C1 — C2

Generators and relations for C52⋊3C42
 G = < a,b,c,d | a5=b5=c4=d4=1, ab=ba, cac-1=a2, dad-1=a-1, cbc-1=b2, bd=db, cd=dc >

Subgroups: 556 in 76 conjugacy classes, 23 normal (11 characteristic)
C1, C2, C2, C4, C22, C5, C5, C2×C4, D5, C10, C10, C42, Dic5, C20, F5, D10, C52, C4×D5, C2×F5, C5⋊D5, C5×C10, C4×F5, C5×Dic5, C5⋊F5, C52⋊C4, C2×C5⋊D5, Dic5⋊2D5, C2×C5⋊F5, C2×C52⋊C4, C52⋊3C42
Quotients: C1, C2, C4, C22, C2×C4, C42, F5, C2×F5, C4×F5, D5⋊F5, C52⋊3C42

Character table of C52⋊3C42

 class 12A2B2C4A4B4C4D4E4F4G4H4I4J4K4L5A5B5C5D10A10B10C10D20A20B20C20D
 size 112525555525252525252525254488448820202020
ρ11111111111111111111111111111    trivial
ρ211111111-1-1-1-1-1-1-1-1111111111111    linear of order 2
ρ31111-1-1-1-11-1-1-1-111111111111-1-1-1-1    linear of order 2
ρ41111-1-1-1-1-11111-1-1-111111111-1-1-1-1    linear of order 2
ρ51-11-1i-i-ii-i-11-11i-ii1111-1-1-1-1-ii-ii    linear of order 4
ρ611-1-1-1-111-i-i-iiiii-i11111111-111-1    linear of order 4
ρ71-1-11i-ii-i-1-iii-i-1111111-1-1-1-1-i-iii    linear of order 4
ρ811-1-1-1-111iii-i-i-i-ii11111111-111-1    linear of order 4
ρ91-11-1i-i-iii1-11-1-ii-i1111-1-1-1-1-ii-ii    linear of order 4
ρ101-1-11i-ii-i1i-i-ii1-1-11111-1-1-1-1-i-iii    linear of order 4
ρ1111-1-111-1-1-iii-i-iii-i111111111-1-11    linear of order 4
ρ121-11-1-iii-i-i1-11-1i-ii1111-1-1-1-1i-ii-i    linear of order 4
ρ131-1-11-ii-ii-1i-i-ii-1111111-1-1-1-1ii-i-i    linear of order 4
ρ141-11-1-iii-ii-11-11-ii-i1111-1-1-1-1i-ii-i    linear of order 4
ρ1511-1-111-1-1i-i-iii-i-ii111111111-1-11    linear of order 4
ρ161-1-11-ii-ii1-iii-i1-1-11111-1-1-1-1ii-i-i    linear of order 4
ρ1744000044000000004-1-1-1-14-1-10-1-10    orthogonal lifted from F5
ρ18440000-4-4000000004-1-1-1-14-1-10110    orthogonal lifted from C2×F5
ρ194400440000000000-14-1-14-1-1-1-100-1    orthogonal lifted from F5
ρ204400-4-40000000000-14-1-14-1-1-11001    orthogonal lifted from C2×F5
ρ214-400004i-4i000000004-1-1-11-4110i-i0    complex lifted from C4×F5
ρ224-400-4i4i0000000000-14-1-1-4111-i00i    complex lifted from C4×F5
ρ234-4004i-4i0000000000-14-1-1-4111i00-i    complex lifted from C4×F5
ρ244-40000-4i4i000000004-1-1-11-4110-ii0    complex lifted from C4×F5
ρ258-800000000000000-2-23-2222-30000    orthogonal faithful
ρ268800000000000000-2-23-2-2-2-230000    orthogonal lifted from D5⋊F5
ρ278800000000000000-2-2-23-2-23-20000    orthogonal lifted from D5⋊F5
ρ288-800000000000000-2-2-2322-320000    orthogonal faithful

Permutation representations of C52⋊3C42
►On 20 points - transitive group 20T91
Generators in S20
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)

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

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

(1 19 6 14)(2 18 7 13)(3 17 8 12)(4 16 9 11)(5 20 10 15)
 
G:=sub<Sym(20)| (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,5,4,3,2)(6,10,9,8,7)(11,12,13,14,15)(16,17,18,19,20), (1,6)(2,9,5,8)(3,7,4,10)(11,20,12,18)(13,16,15,17)(14,19), (1,19,6,14)(2,18,7,13)(3,17,8,12)(4,16,9,11)(5,20,10,15)>;
 
G:=Group( (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,5,4,3,2)(6,10,9,8,7)(11,12,13,14,15)(16,17,18,19,20), (1,6)(2,9,5,8)(3,7,4,10)(11,20,12,18)(13,16,15,17)(14,19), (1,19,6,14)(2,18,7,13)(3,17,8,12)(4,16,9,11)(5,20,10,15) );
 
G=PermutationGroup([[(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,17,18,19,20)], [(1,5,4,3,2),(6,10,9,8,7),(11,12,13,14,15),(16,17,18,19,20)], [(1,6),(2,9,5,8),(3,7,4,10),(11,20,12,18),(13,16,15,17),(14,19)], [(1,19,6,14),(2,18,7,13),(3,17,8,12),(4,16,9,11),(5,20,10,15)]])
 
G:=TransitiveGroup(20,91);
 

Matrix representation of C52⋊3C42 ►in GL8(ℤ)

-1-1-1-10000
10000000
01000000
00100000
00000100
00000010
00000001
0000-1-1-1-1
,
01000000
00100000
00010000
-1-1-1-10000
00000100
00000010
00000001
0000-1-1-1-1
,
10000000
00100000
-1-1-1-10000
01000000
00001000
00000010
0000-1-1-1-1
00000100
,
00001000
00000100
00000010
00000001
-10000000
0-1000000
00-100000
000-10000

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

C52⋊3C42 in GAP, Magma, Sage, TeX 

C_5^2\rtimes_3C_4^2
 
% in TeX
 
G:=Group("C5^2:3C4^2");
 
// GroupNames label
 
G:=SmallGroup(400,124);
 
// by ID
 
G=gap.SmallGroup(400,124);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-2,-5,-5,24,55,1444,970,496,8645,2897]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^5=b^5=c^4=d^4=1,a*b=b*a,c*a*c^-1=a^2,d*a*d^-1=a^-1,c*b*c^-1=b^2,b*d=d*b,c*d=d*c>;
 
// generators/relations
 

Export

Character table of C52⋊3C42 in TeX

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