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G = C52⋊C8order 200 = 23·52

The semidirect product of C52 and C8 acting faithfully

metabelian, soluble, monomial, A-group

Aliases: C52⋊C8, C5⋊D5.C4, C5⋊F5.C2, SmallGroup(200,40)

Series: Derived Chief Lower central Upper central

C1C52 — C52⋊C8
C1C52C5⋊D5C5⋊F5 — C52⋊C8
C52 — C52⋊C8
C1

Generators and relations for C52⋊C8
 G = < a,b,c | a5=b5=c8=1, ab=ba, cac-1=ab2, cbc-1=ab-1 >

25C2
2C5
2C5
2C5
25C4
10D5
10D5
10D5
25C8
10F5
10F5
10F5

Character table of C52⋊C8

 class 124A4B5A5B5C8A8B8C8D
 size 125252588825252525
ρ111111111111    trivial
ρ21111111-1-1-1-1    linear of order 2
ρ311-1-1111i-ii-i    linear of order 4
ρ411-1-1111-ii-ii    linear of order 4
ρ51-1i-i111ζ83ζ8ζ87ζ85    linear of order 8
ρ61-1-ii111ζ8ζ83ζ85ζ87    linear of order 8
ρ71-1-ii111ζ85ζ87ζ8ζ83    linear of order 8
ρ81-1i-i111ζ87ζ85ζ83ζ8    linear of order 8
ρ98000-2-230000    orthogonal faithful
ρ108000-23-20000    orthogonal faithful
ρ1180003-2-20000    orthogonal faithful

Permutation representations of C52⋊C8
On 10 points - transitive group 10T18
Generators in S10
(1 6 8 4 10)(2 5 7 3 9)
(1 6 8 4 10)
(1 2)(3 4 5 6 7 8 9 10)

G:=sub<Sym(10)| (1,6,8,4,10)(2,5,7,3,9), (1,6,8,4,10), (1,2)(3,4,5,6,7,8,9,10)>;

G:=Group( (1,6,8,4,10)(2,5,7,3,9), (1,6,8,4,10), (1,2)(3,4,5,6,7,8,9,10) );

G=PermutationGroup([(1,6,8,4,10),(2,5,7,3,9)], [(1,6,8,4,10)], [(1,2),(3,4,5,6,7,8,9,10)])

G:=TransitiveGroup(10,18);

On 20 points - transitive group 20T56
Generators in S20
(1 15 5 9 19)(2 16 6 10 20)(3 11 17 13 7)(4 12 18 14 8)
(2 16 6 10 20)(4 12 18 14 8)
(1 2 3 4)(5 6 7 8 9 10 11 12)(13 14 15 16 17 18 19 20)

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

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

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

G:=TransitiveGroup(20,56);

On 25 points: primitive - transitive group 25T20
Generators in S25
(1 13 15 11 17)(2 24 3 16 21)(4 10 18 23 5)(6 25 12 7 20)(8 9 19 22 14)
(1 23 25 21 19)(2 22 13 5 12)(3 8 11 10 20)(4 7 24 14 15)(6 16 9 17 18)
(2 3 4 5 6 7 8 9)(10 11 12 13 14 15 16 17)(18 19 20 21 22 23 24 25)

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

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

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

G:=TransitiveGroup(25,20);

C52⋊C8 is a maximal subgroup of   C52⋊M4(2)
C52⋊C8 is a maximal quotient of   C52⋊C16

Polynomial with Galois group C52⋊C8 over ℚ
actionf(x)Disc(f)
10T18x10+2x9-32x8-56x7+288x6+476x5-800x4-1152x3+816x2+704x-416240·36·132·178·434

Matrix representation of C52⋊C8 in GL8(ℤ)

00100000
00010000
-1-1-1-10000
10000000
00000001
0000-1-1-1-1
00001000
00000100
,
01000000
00100000
00010000
-1-1-1-10000
00000010
00000001
0000-1-1-1-1
00001000
,
00001000
00000100
00000010
00000001
10000000
00010000
01000000
-1-1-1-10000

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

C52⋊C8 in GAP, Magma, Sage, TeX

C_5^2\rtimes C_8
% in TeX

G:=Group("C5^2:C8");
// GroupNames label

G:=SmallGroup(200,40);
// by ID

G=gap.SmallGroup(200,40);
# by ID

G:=PCGroup([5,-2,-2,-2,-5,5,10,26,3523,168,173,3404,1009,1014]);
// Polycyclic

G:=Group<a,b,c|a^5=b^5=c^8=1,a*b=b*a,c*a*c^-1=a*b^2,c*b*c^-1=a*b^-1>;
// generators/relations

Export

Subgroup lattice of C52⋊C8 in TeX
Character table of C52⋊C8 in TeX

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