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G = C34⋊C5order 405 = 34·5

The semidirect product of C34 and C5 acting faithfully

metabelian, soluble, monomial, A-group

Aliases: C34⋊C5, SmallGroup(405,15)

Series: Derived Chief Lower central Upper central

C1C34 — C34⋊C5
C1C34 — C34⋊C5
C34 — C34⋊C5
C1

Generators and relations for C34⋊C5
 G = < a,b,c,d,e | a3=b3=c3=d3=e5=1, ab=ba, ac=ca, ad=da, eae-1=a-1b-1c, bc=cb, bd=db, ebe-1=a, cd=dc, ece-1=a-1bd-1, ede-1=b-1c-1 >

Subgroups: 294 in 46 conjugacy classes, 3 normal (all characteristic)
C1, C3 [×8], C5, C32 [×26], C33 [×8], C34, C34⋊C5
Quotients: C1, C5, C34⋊C5

Character table of C34⋊C5

 class 13A3B3C3D3E3F3G3H3I3J3K3L3M3N3O3P5A5B5C5D
 size 1555555555555555581818181
ρ1111111111111111111111    trivial
ρ211111111111111111ζ5ζ52ζ53ζ54    linear of order 5
ρ311111111111111111ζ53ζ5ζ54ζ52    linear of order 5
ρ411111111111111111ζ52ζ54ζ5ζ53    linear of order 5
ρ511111111111111111ζ54ζ53ζ52ζ5    linear of order 5
ρ65-5-3-3/21-3-3/22-11-3-3/221+3-3/2-1-5+3-3/21+3-3/2-12-12-1-10000    complex faithful
ρ751-3-3/2-5+3-3/2-12-1-1-121+3-3/2-5-3-3/21-3-3/221+3-3/22-1-10000    complex faithful
ρ851-3-3/2-12-1-5+3-3/22-5-3-3/2-11+3-3/2-11+3-3/2-11-3-3/2-1220000    complex faithful
ρ951+3-3/2-5-3-3/2-12-1-1-121-3-3/2-5+3-3/21+3-3/221-3-3/22-1-10000    complex faithful
ρ105-12-5-3-3/2-1-1-5+3-3/2-1-1-1221+3-3/221-3-3/21+3-3/21-3-3/20000    complex faithful
ρ115-11+3-3/2-121-3-3/2-11+3-3/22-11-3-3/2-5+3-3/2-1-5-3-3/2-1220000    complex faithful
ρ125221-3-3/21-3-3/2-11+3-3/2-11+3-3/222-1-1-1-1-5-3-3/2-5+3-3/20000    complex faithful
ρ135-1-11-3-3/21+3-3/221+3-3/221-3-3/2-1-12-5-3-3/22-5+3-3/2-1-10000    complex faithful
ρ145-5+3-3/21+3-3/22-11+3-3/221-3-3/2-1-5-3-3/21-3-3/2-12-12-1-10000    complex faithful
ρ1551+3-3/2-12-1-5-3-3/22-5+3-3/2-11-3-3/2-11-3-3/2-11+3-3/2-1220000    complex faithful
ρ165-12-5+3-3/2-1-1-5-3-3/2-1-1-1221-3-3/221+3-3/21-3-3/21+3-3/20000    complex faithful
ρ1752-1-1-5+3-3/22-12-5-3-3/22-1-11+3-3/2-11-3-3/21-3-3/21+3-3/20000    complex faithful
ρ1852-1-1-5-3-3/22-12-5+3-3/22-1-11-3-3/2-11+3-3/21+3-3/21-3-3/20000    complex faithful
ρ195-11-3-3/2-121+3-3/2-11-3-3/22-11+3-3/2-5-3-3/2-1-5+3-3/2-1220000    complex faithful
ρ205-1-11+3-3/21-3-3/221-3-3/221+3-3/2-1-12-5+3-3/22-5-3-3/2-1-10000    complex faithful
ρ215221+3-3/21+3-3/2-11-3-3/2-11-3-3/222-1-1-1-1-5+3-3/2-5-3-3/20000    complex faithful

Permutation representations of C34⋊C5
On 15 points - transitive group 15T26
Generators in S15
(1 8 12)(2 13 9)(3 10 14)(4 6 15)(5 7 11)
(1 8 12)(2 9 13)(3 14 10)(4 6 15)(5 7 11)
(1 8 12)(2 9 13)(3 10 14)
(1 12 8)(2 9 13)(3 10 14)(5 11 7)
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)

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

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

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

G:=TransitiveGroup(15,26);

Polynomial with Galois group C34⋊C5 over ℚ
actionf(x)Disc(f)
15T26x15-150x13-520x12+2400x11+12366x10-1700x9-73410x8-60675x7+161150x6+214578x5-119280x4-247825x3-3750x2+93525x+23255320·524·716·432·1512·34572·54072·159438609894012

Matrix representation of C34⋊C5 in GL5(𝔽31)

50000
05000
00500
00050
4303025
,
50000
05000
00500
12232250
00005
,
50000
01000
00100
0171950
061305
,
250000
025000
00100
1981950
2711305
,
01000
00100
2921240
000101
000150

G:=sub<GL(5,GF(31))| [5,0,0,0,4,0,5,0,0,30,0,0,5,0,3,0,0,0,5,0,0,0,0,0,25],[5,0,0,12,0,0,5,0,23,0,0,0,5,2,0,0,0,0,25,0,0,0,0,0,5],[5,0,0,0,0,0,1,0,17,6,0,0,1,19,13,0,0,0,5,0,0,0,0,0,5],[25,0,0,19,27,0,25,0,8,1,0,0,1,19,13,0,0,0,5,0,0,0,0,0,5],[0,0,2,0,0,1,0,9,0,0,0,1,21,0,0,0,0,24,10,15,0,0,0,1,0] >;

C34⋊C5 in GAP, Magma, Sage, TeX

C_3^4\rtimes C_5
% in TeX

G:=Group("C3^4:C5");
// GroupNames label

G:=SmallGroup(405,15);
// by ID

G=gap.SmallGroup(405,15);
# by ID

G:=PCGroup([5,-5,-3,3,3,3,3751,827,3303,9129]);
// Polycyclic

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

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Character table of C34⋊C5 in TeX

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