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G = C52⋊C6order 150 = 2·3·52

The semidirect product of C52 and C6 acting faithfully

metabelian, soluble, monomial, A-group

Aliases: C52⋊C6, C5⋊D5⋊C3, C52⋊C32C2, SmallGroup(150,6)

Series: Derived Chief Lower central Upper central

C1C52 — C52⋊C6
C1C52C52⋊C3 — C52⋊C6
C52 — C52⋊C6
C1

Generators and relations for C52⋊C6
 G = < a,b,c | a5=b5=c6=1, ab=ba, cac-1=a2b3, cbc-1=a-1b-1 >

25C2
25C3
3C5
3C5
25C6
15D5
15D5

Character table of C52⋊C6

 class 123A3B5A5B5C5D6A6B
 size 125252566662525
ρ11111111111    trivial
ρ21-1111111-1-1    linear of order 2
ρ311ζ32ζ31111ζ32ζ3    linear of order 3
ρ41-1ζ32ζ31111ζ6ζ65    linear of order 6
ρ51-1ζ3ζ321111ζ65ζ6    linear of order 6
ρ611ζ3ζ321111ζ3ζ32    linear of order 3
ρ76000-3-5/21-51+5-3+5/200    orthogonal faithful
ρ860001+5-3-5/2-3+5/21-500    orthogonal faithful
ρ96000-3+5/21+51-5-3-5/200    orthogonal faithful
ρ1060001-5-3+5/2-3-5/21+500    orthogonal faithful

Permutation representations of C52⋊C6
On 15 points - transitive group 15T12
Generators in S15
(1 8 11 14 5)(2 12 6 9 15)(3 4 13 10 7)
(2 6 15 12 9)(3 7 10 13 4)
(1 2 3)(4 5 6 7 8 9)(10 11 12 13 14 15)

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

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

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

G:=TransitiveGroup(15,12);

On 25 points: primitive - transitive group 25T15
Generators in S25
(1 21 5 2 24)(3 14 4 11 13)(6 10 8 7 17)(9 19 25 20 15)(12 18 23 22 16)
(1 11 16 19 8)(2 14 23 15 6)(3 18 20 17 5)(4 22 9 10 24)(7 21 13 12 25)
(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)| (1,21,5,2,24)(3,14,4,11,13)(6,10,8,7,17)(9,19,25,20,15)(12,18,23,22,16), (1,11,16,19,8)(2,14,23,15,6)(3,18,20,17,5)(4,22,9,10,24)(7,21,13,12,25), (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( (1,21,5,2,24)(3,14,4,11,13)(6,10,8,7,17)(9,19,25,20,15)(12,18,23,22,16), (1,11,16,19,8)(2,14,23,15,6)(3,18,20,17,5)(4,22,9,10,24)(7,21,13,12,25), (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([(1,21,5,2,24),(3,14,4,11,13),(6,10,8,7,17),(9,19,25,20,15),(12,18,23,22,16)], [(1,11,16,19,8),(2,14,23,15,6),(3,18,20,17,5),(4,22,9,10,24),(7,21,13,12,25)], [(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(25,15);

On 30 points - transitive group 30T35
Generators in S30
(1 22 25 8 14)(2 26 15 23 9)(3 24 27 10 16)(4 17 11 28 19)(5 12 20 18 29)(6 13 7 30 21)
(2 15 9 26 23)(3 16 10 27 24)(5 20 29 12 18)(6 21 30 7 13)
(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)

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

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

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

G:=TransitiveGroup(30,35);

C52⋊C6 is a maximal subgroup of   C52⋊Dic3  C52⋊C12  C52⋊D6  C5⋊D15⋊C3
C52⋊C6 is a maximal quotient of   C522C12  C52⋊C18  C5⋊D15⋊C3

Polynomial with Galois group C52⋊C6 over ℚ
actionf(x)Disc(f)
15T12x15-235x13+70x12+15930x11-14493x10-325950x9+112750x8+2876560x7+708890x6-11702794x5-8278925x4+19171805x3+20087950x2-6362585x-9179941312·518·722·132·4118·22672·48712·5483031859935751932

Matrix representation of C52⋊C6 in GL6(𝔽31)

100000
010000
131301800
3019131300
3019001313
131001830
,
3010000
11190000
3000100
121301800
2919001313
121001830
,
131001730
3019001213
0000300
12100300
0010300
1311830300

G:=sub<GL(6,GF(31))| [1,0,13,30,30,13,0,1,1,19,19,1,0,0,30,13,0,0,0,0,18,13,0,0,0,0,0,0,13,18,0,0,0,0,13,30],[30,11,30,12,29,12,1,19,0,1,19,1,0,0,0,30,0,0,0,0,1,18,0,0,0,0,0,0,13,18,0,0,0,0,13,30],[13,30,0,12,0,13,1,19,0,1,0,1,0,0,0,0,1,18,0,0,0,0,0,30,17,12,30,30,30,30,30,13,0,0,0,0] >;

C52⋊C6 in GAP, Magma, Sage, TeX

C_5^2\rtimes C_6
% in TeX

G:=Group("C5^2:C6");
// GroupNames label

G:=SmallGroup(150,6);
// by ID

G=gap.SmallGroup(150,6);
# by ID

G:=PCGroup([4,-2,-3,-5,5,290,474,1923,295]);
// Polycyclic

G:=Group<a,b,c|a^5=b^5=c^6=1,a*b=b*a,c*a*c^-1=a^2*b^3,c*b*c^-1=a^-1*b^-1>;
// generators/relations

Export

Subgroup lattice of C52⋊C6 in TeX
Character table of C52⋊C6 in TeX

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