metabelian, soluble, monomial, A-group
Aliases: C34⋊C5, SmallGroup(405,15)
Series: Derived ►Chief ►Lower central ►Upper central
| C34 — C34⋊C5 |
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, C5, C32, C33, C34, C34⋊C5
Quotients: C1, C5, C34⋊C5
Character table of C34⋊C5
| class | 1 | 3A | 3B | 3C | 3D | 3E | 3F | 3G | 3H | 3I | 3J | 3K | 3L | 3M | 3N | 3O | 3P | 5A | 5B | 5C | 5D | |
| size | 1 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 81 | 81 | 81 | 81 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ5 | ζ52 | ζ53 | ζ54 | linear of order 5 |
| ρ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ53 | ζ5 | ζ54 | ζ52 | linear of order 5 |
| ρ4 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ52 | ζ54 | ζ5 | ζ53 | linear of order 5 |
| ρ5 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ54 | ζ53 | ζ52 | ζ5 | linear of order 5 |
| ρ6 | 5 | -5-3√-3/2 | 1-3√-3/2 | 2 | -1 | 1-3√-3/2 | 2 | 1+3√-3/2 | -1 | -5+3√-3/2 | 1+3√-3/2 | -1 | 2 | -1 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | complex faithful |
| ρ7 | 5 | 1-3√-3/2 | -5+3√-3/2 | -1 | 2 | -1 | -1 | -1 | 2 | 1+3√-3/2 | -5-3√-3/2 | 1-3√-3/2 | 2 | 1+3√-3/2 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | complex faithful |
| ρ8 | 5 | 1-3√-3/2 | -1 | 2 | -1 | -5+3√-3/2 | 2 | -5-3√-3/2 | -1 | 1+3√-3/2 | -1 | 1+3√-3/2 | -1 | 1-3√-3/2 | -1 | 2 | 2 | 0 | 0 | 0 | 0 | complex faithful |
| ρ9 | 5 | 1+3√-3/2 | -5-3√-3/2 | -1 | 2 | -1 | -1 | -1 | 2 | 1-3√-3/2 | -5+3√-3/2 | 1+3√-3/2 | 2 | 1-3√-3/2 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | complex faithful |
| ρ10 | 5 | -1 | 2 | -5-3√-3/2 | -1 | -1 | -5+3√-3/2 | -1 | -1 | -1 | 2 | 2 | 1+3√-3/2 | 2 | 1-3√-3/2 | 1+3√-3/2 | 1-3√-3/2 | 0 | 0 | 0 | 0 | complex faithful |
| ρ11 | 5 | -1 | 1+3√-3/2 | -1 | 2 | 1-3√-3/2 | -1 | 1+3√-3/2 | 2 | -1 | 1-3√-3/2 | -5+3√-3/2 | -1 | -5-3√-3/2 | -1 | 2 | 2 | 0 | 0 | 0 | 0 | complex faithful |
| ρ12 | 5 | 2 | 2 | 1-3√-3/2 | 1-3√-3/2 | -1 | 1+3√-3/2 | -1 | 1+3√-3/2 | 2 | 2 | -1 | -1 | -1 | -1 | -5-3√-3/2 | -5+3√-3/2 | 0 | 0 | 0 | 0 | complex faithful |
| ρ13 | 5 | -1 | -1 | 1-3√-3/2 | 1+3√-3/2 | 2 | 1+3√-3/2 | 2 | 1-3√-3/2 | -1 | -1 | 2 | -5-3√-3/2 | 2 | -5+3√-3/2 | -1 | -1 | 0 | 0 | 0 | 0 | complex faithful |
| ρ14 | 5 | -5+3√-3/2 | 1+3√-3/2 | 2 | -1 | 1+3√-3/2 | 2 | 1-3√-3/2 | -1 | -5-3√-3/2 | 1-3√-3/2 | -1 | 2 | -1 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | complex faithful |
| ρ15 | 5 | 1+3√-3/2 | -1 | 2 | -1 | -5-3√-3/2 | 2 | -5+3√-3/2 | -1 | 1-3√-3/2 | -1 | 1-3√-3/2 | -1 | 1+3√-3/2 | -1 | 2 | 2 | 0 | 0 | 0 | 0 | complex faithful |
| ρ16 | 5 | -1 | 2 | -5+3√-3/2 | -1 | -1 | -5-3√-3/2 | -1 | -1 | -1 | 2 | 2 | 1-3√-3/2 | 2 | 1+3√-3/2 | 1-3√-3/2 | 1+3√-3/2 | 0 | 0 | 0 | 0 | complex faithful |
| ρ17 | 5 | 2 | -1 | -1 | -5+3√-3/2 | 2 | -1 | 2 | -5-3√-3/2 | 2 | -1 | -1 | 1+3√-3/2 | -1 | 1-3√-3/2 | 1-3√-3/2 | 1+3√-3/2 | 0 | 0 | 0 | 0 | complex faithful |
| ρ18 | 5 | 2 | -1 | -1 | -5-3√-3/2 | 2 | -1 | 2 | -5+3√-3/2 | 2 | -1 | -1 | 1-3√-3/2 | -1 | 1+3√-3/2 | 1+3√-3/2 | 1-3√-3/2 | 0 | 0 | 0 | 0 | complex faithful |
| ρ19 | 5 | -1 | 1-3√-3/2 | -1 | 2 | 1+3√-3/2 | -1 | 1-3√-3/2 | 2 | -1 | 1+3√-3/2 | -5-3√-3/2 | -1 | -5+3√-3/2 | -1 | 2 | 2 | 0 | 0 | 0 | 0 | complex faithful |
| ρ20 | 5 | -1 | -1 | 1+3√-3/2 | 1-3√-3/2 | 2 | 1-3√-3/2 | 2 | 1+3√-3/2 | -1 | -1 | 2 | -5+3√-3/2 | 2 | -5-3√-3/2 | -1 | -1 | 0 | 0 | 0 | 0 | complex faithful |
| ρ21 | 5 | 2 | 2 | 1+3√-3/2 | 1+3√-3/2 | -1 | 1-3√-3/2 | -1 | 1-3√-3/2 | 2 | 2 | -1 | -1 | -1 | -1 | -5+3√-3/2 | -5-3√-3/2 | 0 | 0 | 0 | 0 | complex faithful |
►On 15 points - transitive group 15T26
Generators in S15
(1 9 12)(2 13 10)(3 6 14)(4 7 15)(5 8 11)
(1 9 12)(2 10 13)(3 14 6)(4 7 15)(5 8 11)
(1 9 12)(2 10 13)(3 6 14)
(1 12 9)(2 10 13)(3 6 14)(5 11 8)
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)
G:=sub<Sym(15)| (1,9,12)(2,13,10)(3,6,14)(4,7,15)(5,8,11), (1,9,12)(2,10,13)(3,14,6)(4,7,15)(5,8,11), (1,9,12)(2,10,13)(3,6,14), (1,12,9)(2,10,13)(3,6,14)(5,11,8), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)>;
G:=Group( (1,9,12)(2,13,10)(3,6,14)(4,7,15)(5,8,11), (1,9,12)(2,10,13)(3,14,6)(4,7,15)(5,8,11), (1,9,12)(2,10,13)(3,6,14), (1,12,9)(2,10,13)(3,6,14)(5,11,8), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15) );
G=PermutationGroup([[(1,9,12),(2,13,10),(3,6,14),(4,7,15),(5,8,11)], [(1,9,12),(2,10,13),(3,14,6),(4,7,15),(5,8,11)], [(1,9,12),(2,10,13),(3,6,14)], [(1,12,9),(2,10,13),(3,6,14),(5,11,8)], [(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
| action | f(x) | Disc(f) |
|---|---|---|
| 15T26 | x15-150x13-520x12+2400x11+12366x10-1700x9-73410x8-60675x7+161150x6+214578x5-119280x4-247825x3-3750x2+93525x+23255 | 320·524·716·432·1512·34572·54072·159438609894012 |
Matrix representation of C34⋊C5 ►in GL5(𝔽31)
| 5 | 0 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 | 0 |
| 0 | 0 | 5 | 0 | 0 |
| 0 | 0 | 0 | 5 | 0 |
| 4 | 30 | 3 | 0 | 25 |
| 5 | 0 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 | 0 |
| 0 | 0 | 5 | 0 | 0 |
| 12 | 23 | 2 | 25 | 0 |
| 0 | 0 | 0 | 0 | 5 |
| 5 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 17 | 19 | 5 | 0 |
| 0 | 6 | 13 | 0 | 5 |
| 25 | 0 | 0 | 0 | 0 |
| 0 | 25 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 19 | 8 | 19 | 5 | 0 |
| 27 | 1 | 13 | 0 | 5 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 2 | 9 | 21 | 24 | 0 |
| 0 | 0 | 0 | 10 | 1 |
| 0 | 0 | 0 | 15 | 0 |
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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