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G = C31⋊C15  order 465 = 3·5·31

The semidirect product of C31 and C15 acting faithfully

metacyclic, supersoluble, monomial, Z-group

Aliases: C31⋊C15, C31⋊C5⋊C3, C31⋊C3⋊C5, SmallGroup(465,1)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C31 — C31⋊C15
Chief series
C1 — C31 — C31⋊C5 — C31⋊C15
Lower central
C31 — C31⋊C15
Upper central
C1

Generators and relations for C31⋊C15
 G = < a,b | a31=b15=1, bab-1=a20 >

C1
31C3
31C5
C31
31C15
C31⋊C3
C31⋊C5
C31⋊C15

Character table of C31⋊C15

 class 13A3B5A5B5C5D15A15B15C15D15E15F15G15H31A31B
 size 131313131313131313131313131311515
ρ111111111111111111    trivial
ρ21ζ32ζ31111ζ32ζ3ζ3ζ3ζ32ζ32ζ32ζ311    linear of order 3
ρ31ζ3ζ321111ζ3ζ32ζ32ζ32ζ3ζ3ζ3ζ3211    linear of order 3
ρ4111ζ5ζ53ζ52ζ54ζ53ζ5ζ53ζ52ζ52ζ54ζ5ζ5411    linear of order 5
ρ5111ζ53ζ54ζ5ζ52ζ54ζ53ζ54ζ5ζ5ζ52ζ53ζ5211    linear of order 5
ρ6111ζ54ζ52ζ53ζ5ζ52ζ54ζ52ζ53ζ53ζ5ζ54ζ511    linear of order 5
ρ7111ζ52ζ5ζ54ζ53ζ5ζ52ζ5ζ54ζ54ζ53ζ52ζ5311    linear of order 5
ρ81ζ32ζ3ζ53ζ54ζ5ζ52ζ32ζ54ζ3ζ53ζ3ζ54ζ3ζ5ζ32ζ5ζ32ζ52ζ32ζ53ζ3ζ5211    linear of order 15
ρ91ζ3ζ32ζ54ζ52ζ53ζ5ζ3ζ52ζ32ζ54ζ32ζ52ζ32ζ53ζ3ζ53ζ3ζ5ζ3ζ54ζ32ζ511    linear of order 15
ρ101ζ32ζ3ζ5ζ53ζ52ζ54ζ32ζ53ζ3ζ5ζ3ζ53ζ3ζ52ζ32ζ52ζ32ζ54ζ32ζ5ζ3ζ5411    linear of order 15
ρ111ζ32ζ3ζ52ζ5ζ54ζ53ζ32ζ5ζ3ζ52ζ3ζ5ζ3ζ54ζ32ζ54ζ32ζ53ζ32ζ52ζ3ζ5311    linear of order 15
ρ121ζ3ζ32ζ53ζ54ζ5ζ52ζ3ζ54ζ32ζ53ζ32ζ54ζ32ζ5ζ3ζ5ζ3ζ52ζ3ζ53ζ32ζ5211    linear of order 15
ρ131ζ32ζ3ζ54ζ52ζ53ζ5ζ32ζ52ζ3ζ54ζ3ζ52ζ3ζ53ζ32ζ53ζ32ζ5ζ32ζ54ζ3ζ511    linear of order 15
ρ141ζ3ζ32ζ52ζ5ζ54ζ53ζ3ζ5ζ32ζ52ζ32ζ5ζ32ζ54ζ3ζ54ζ3ζ53ζ3ζ52ζ32ζ5311    linear of order 15
ρ151ζ3ζ32ζ5ζ53ζ52ζ54ζ3ζ53ζ32ζ5ζ32ζ53ζ32ζ52ζ3ζ52ζ3ζ54ζ3ζ5ζ32ζ5411    linear of order 15
ρ161500000000000000-1+√-31/2-1-√-31/2    complex faithful
ρ171500000000000000-1-√-31/2-1+√-31/2    complex faithful

Permutation representations of C31⋊C15
►On 31 points: primitive - transitive group 31T7
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 15 11 17 8 6 9 20 19 5 26 10 3 29 21)(4 12 31 18 22 16 25 27 24 13 14 28 7 23 30)
 
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), (2,15,11,17,8,6,9,20,19,5,26,10,3,29,21)(4,12,31,18,22,16,25,27,24,13,14,28,7,23,30)>;
 
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), (2,15,11,17,8,6,9,20,19,5,26,10,3,29,21)(4,12,31,18,22,16,25,27,24,13,14,28,7,23,30) );
 
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)], [(2,15,11,17,8,6,9,20,19,5,26,10,3,29,21),(4,12,31,18,22,16,25,27,24,13,14,28,7,23,30)]])
 
G:=TransitiveGroup(31,7);
 

Matrix representation of C31⋊C15 ►in GL15(𝔽1861)

010000000000000
001000000000000
000100000000000
000010000000000
000001000000000
000000100000000
000000010000000
000000001000000
000000000100000
000000000010000
000000000001000
000000000000100
000000000000010
000000000000001
187487098211548729859908751857113993877873
,
100000000000000
175075218441324111276597722911173922035417631637868
000000000100000
1857110232127631855118881174411399716407541854987
112111217674207432231759640974124152515002091214994
000000010000000
8806441614323348175786012101261628108514461347701745
87318541096235883174611199517511855110512163717431857
000001000000000
98223213015311625106111634863110777052618441094117
000000000000001
000100000000000
5761162598227917401071107760185033611138831746
000000000000100
010000000000000

G:=sub<GL(15,GF(1861))| [0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,874,0,1,0,0,0,0,0,0,0,0,0,0,0,0,870,0,0,1,0,0,0,0,0,0,0,0,0,0,0,982,0,0,0,1,0,0,0,0,0,0,0,0,0,0,115,0,0,0,0,1,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,0,0,0,872,0,0,0,0,0,0,1,0,0,0,0,0,0,0,985,0,0,0,0,0,0,0,1,0,0,0,0,0,0,990,0,0,0,0,0,0,0,0,1,0,0,0,0,0,875,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1857,0,0,0,0,0,0,0,0,0,0,1,0,0,0,113,0,0,0,0,0,0,0,0,0,0,0,1,0,0,993,0,0,0,0,0,0,0,0,0,0,0,0,1,0,877,0,0,0,0,0,0,0,0,0,0,0,0,0,1,873],[1,1750,0,1857,112,0,880,873,0,982,0,0,5,0,0,0,752,0,110,1112,0,644,1854,0,232,0,0,761,0,1,0,1844,0,232,1767,0,1614,1096,0,130,0,0,1625,0,0,0,1324,0,12,420,0,323,235,0,1531,0,1,98,0,0,0,1112,0,763,743,0,348,883,0,1625,0,0,227,0,0,0,765,0,1855,223,0,1757,1746,1,106,0,0,9,0,0,0,977,0,118,1759,0,860,111,0,11,0,0,1740,0,0,0,229,0,881,640,1,1210,995,0,1634,0,0,107,0,0,0,11,0,1744,974,0,126,1751,0,863,0,0,1107,0,0,0,1739,1,113,124,0,1628,1855,0,1107,0,0,760,0,0,0,220,0,997,1525,0,1085,1105,0,770,0,0,1850,0,0,0,354,0,1640,1500,0,1446,12,0,526,0,0,336,0,0,0,1763,0,754,209,0,134,1637,0,1844,0,0,1113,1,0,0,1637,0,1854,1214,0,770,1743,0,1094,0,0,883,0,0,0,868,0,987,994,0,1745,1857,0,117,1,0,1746,0,0] >;
 

C31⋊C15 in GAP, Magma, Sage, TeX 

C_{31}\rtimes C_{15}
 
% in TeX
 
G:=Group("C31:C15");
 
// GroupNames label
 
G:=SmallGroup(465,1);
 
// by ID
 
G=gap.SmallGroup(465,1);
 
# by ID
 
G:=PCGroup([3,-3,-5,-31,3377,725]);
 
// Polycyclic
 
G:=Group<a,b|a^31=b^15=1,b*a*b^-1=a^20>;
 
// generators/relations
 

Export

Subgroup lattice of C31⋊C15 in TeX
Character table of C31⋊C15 in TeX

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