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G = C31⋊C10order 310 = 2·5·31

The semidirect product of C31 and C10 acting faithfully

metacyclic, supersoluble, monomial, Z-group

Aliases: C31⋊C10, D31⋊C5, C31⋊C5⋊C2, SmallGroup(310,1)

Series: Derived Chief Lower central Upper central

C1C31 — C31⋊C10
C1C31C31⋊C5 — C31⋊C10
C31 — C31⋊C10
C1

Generators and relations for C31⋊C10
 G = < a,b | a31=b10=1, bab-1=a23 >

31C2
31C5
31C10

Character table of C31⋊C10

 class 125A5B5C5D10A10B10C10D31A31B31C
 size 1313131313131313131101010
ρ11111111111111    trivial
ρ21-11111-1-1-1-1111    linear of order 2
ρ311ζ52ζ54ζ5ζ53ζ5ζ54ζ53ζ52111    linear of order 5
ρ411ζ5ζ52ζ53ζ54ζ53ζ52ζ54ζ5111    linear of order 5
ρ51-1ζ54ζ53ζ52ζ55253554111    linear of order 10
ρ61-1ζ5ζ52ζ53ζ545352545111    linear of order 10
ρ711ζ54ζ53ζ52ζ5ζ52ζ53ζ5ζ54111    linear of order 5
ρ81-1ζ52ζ54ζ5ζ535545352111    linear of order 10
ρ91-1ζ53ζ5ζ54ζ525455253111    linear of order 10
ρ1011ζ53ζ5ζ54ζ52ζ54ζ5ζ52ζ53111    linear of order 5
ρ1110000000000ζ3128312531243119311731143112317316313ζ31303129312731233116311531831431231ζ31263122312131203118311331113110319315    orthogonal faithful
ρ1210000000000ζ31303129312731233116311531831431231ζ31263122312131203118311331113110319315ζ3128312531243119311731143112317316313    orthogonal faithful
ρ1310000000000ζ31263122312131203118311331113110319315ζ3128312531243119311731143112317316313ζ31303129312731233116311531831431231    orthogonal faithful

Permutation representations of C31⋊C10
On 31 points: primitive - transitive group 31T6
Generators in S31
(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)
(2 28 17 30 9 31 5 16 3 24)(4 20 18 26 25 29 13 15 7 8)(6 12 19 22 10 27 21 14 11 23)

G:=sub<Sym(31)| (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), (2,28,17,30,9,31,5,16,3,24)(4,20,18,26,25,29,13,15,7,8)(6,12,19,22,10,27,21,14,11,23)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31), (2,28,17,30,9,31,5,16,3,24)(4,20,18,26,25,29,13,15,7,8)(6,12,19,22,10,27,21,14,11,23) );

G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31)], [(2,28,17,30,9,31,5,16,3,24),(4,20,18,26,25,29,13,15,7,8),(6,12,19,22,10,27,21,14,11,23)])

G:=TransitiveGroup(31,6);

Matrix representation of C31⋊C10 in GL10(𝔽2)

0000101011
0001100001
0101111100
0011100001
0010010001
1011000100
0011110101
0000100100
0001111000
0011101101
,
1010110100
0110011000
0100111001
0110010101
0010000000
0100001100
0110010001
0000010001
0101111100
0110110110

G:=sub<GL(10,GF(2))| [0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,1,1,1,0,0,1,0,1,1,1,0,1,1,0,1,1,1,1,1,1,0,0,1,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0,1,1,0,0,1,0,0,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,1,1,0,1,1,0,1,0,0,1],[1,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,1,0,1,1,1,1,0,1,1,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,1,0,1,0,0,0,0,0,1,1,1,1,1,1,0,0,1,1,1,1,0,1,1,0,0,1,0,0,1,0,1,0,0,1,0,1,0,0,1,1,0,0,0,0,0,0,0,0,0,1,0,0,1,1,0,0,1,1,0,0] >;

C31⋊C10 in GAP, Magma, Sage, TeX

C_{31}\rtimes C_{10}
% in TeX

G:=Group("C31:C10");
// GroupNames label

G:=SmallGroup(310,1);
// by ID

G=gap.SmallGroup(310,1);
# by ID

G:=PCGroup([3,-2,-5,-31,2702,725]);
// Polycyclic

G:=Group<a,b|a^31=b^10=1,b*a*b^-1=a^23>;
// generators/relations

Export

Subgroup lattice of C31⋊C10 in TeX
Character table of C31⋊C10 in TeX

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