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G = C52⋊C12order 300 = 22·3·52

The semidirect product of C52 and C12 acting faithfully

metabelian, soluble, monomial, A-group

Aliases: C52⋊C12, C5⋊D5.C6, C5⋊F5⋊C3, C52⋊C32C4, C52⋊C6.1C2, SmallGroup(300,24)

Series: Derived Chief Lower central Upper central

C1C52 — C52⋊C12
C1C52C5⋊D5C52⋊C6 — C52⋊C12
C52 — C52⋊C12
C1

Generators and relations for C52⋊C12
 G = < a,b,c | a5=b5=c12=1, ab=ba, cac-1=a-1b, cbc-1=a3b3 >

25C2
25C3
3C5
3C5
25C4
25C6
15D5
15D5
25C12
15F5
15F5

Character table of C52⋊C12

 class 123A3B4A4B5A5B6A6B12A12B12C12D
 size 125252525251212252525252525
ρ111111111111111    trivial
ρ21111-1-11111-1-1-1-1    linear of order 2
ρ311ζ32ζ3-1-111ζ3ζ32ζ6ζ6ζ65ζ65    linear of order 6
ρ411ζ3ζ321111ζ32ζ3ζ3ζ3ζ32ζ32    linear of order 3
ρ511ζ3ζ32-1-111ζ32ζ3ζ65ζ65ζ6ζ6    linear of order 6
ρ611ζ32ζ31111ζ3ζ32ζ32ζ32ζ3ζ3    linear of order 3
ρ71-111-ii11-1-1i-ii-i    linear of order 4
ρ81-111i-i11-1-1-ii-ii    linear of order 4
ρ91-1ζ3ζ32-ii11ζ6ζ65ζ4ζ3ζ43ζ3ζ4ζ32ζ43ζ32    linear of order 12
ρ101-1ζ32ζ3-ii11ζ65ζ6ζ4ζ32ζ43ζ32ζ4ζ3ζ43ζ3    linear of order 12
ρ111-1ζ3ζ32i-i11ζ6ζ65ζ43ζ3ζ4ζ3ζ43ζ32ζ4ζ32    linear of order 12
ρ121-1ζ32ζ3i-i11ζ65ζ6ζ43ζ32ζ4ζ32ζ43ζ3ζ4ζ3    linear of order 12
ρ1312000002-3000000    orthogonal faithful
ρ141200000-32000000    orthogonal faithful

Permutation representations of C52⋊C12
On 15 points - transitive group 15T19
Generators in S15
(1 12 9 15 6)(2 13 10 4 7)(3 5 14 8 11)
(1 9 6 12 15)(2 7 4 10 13)
(1 2 3)(4 5 6 7 8 9 10 11 12 13 14 15)

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

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

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

G:=TransitiveGroup(15,19);

On 25 points: primitive - transitive group 25T26
Generators in S25
(1 15 24 18 21)(2 9 23 25 13)(3 8 7 19 17)(4 12 16 5 14)(6 10 20 11 22)
(1 4 13 7 10)(2 19 20 15 12)(3 22 18 5 23)(6 21 14 25 8)(9 17 11 24 16)
(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)| (1,15,24,18,21)(2,9,23,25,13)(3,8,7,19,17)(4,12,16,5,14)(6,10,20,11,22), (1,4,13,7,10)(2,19,20,15,12)(3,22,18,5,23)(6,21,14,25,8)(9,17,11,24,16), (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( (1,15,24,18,21)(2,9,23,25,13)(3,8,7,19,17)(4,12,16,5,14)(6,10,20,11,22), (1,4,13,7,10)(2,19,20,15,12)(3,22,18,5,23)(6,21,14,25,8)(9,17,11,24,16), (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([(1,15,24,18,21),(2,9,23,25,13),(3,8,7,19,17),(4,12,16,5,14),(6,10,20,11,22)], [(1,4,13,7,10),(2,19,20,15,12),(3,22,18,5,23),(6,21,14,25,8),(9,17,11,24,16)], [(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(25,26);

On 30 points - transitive group 30T78
Generators in S30
(1 22 9 15 28)(3 24 11 17 30)(4 12 19 25 18)(6 14 21 27 8)
(1 22 9 15 28)(2 23 10 16 29)(3 17 24 30 11)(4 12 19 25 18)(5 13 20 26 7)(6 27 14 8 21)
(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)| (1,22,9,15,28)(3,24,11,17,30)(4,12,19,25,18)(6,14,21,27,8), (1,22,9,15,28)(2,23,10,16,29)(3,17,24,30,11)(4,12,19,25,18)(5,13,20,26,7)(6,27,14,8,21), (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( (1,22,9,15,28)(3,24,11,17,30)(4,12,19,25,18)(6,14,21,27,8), (1,22,9,15,28)(2,23,10,16,29)(3,17,24,30,11)(4,12,19,25,18)(5,13,20,26,7)(6,27,14,8,21), (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([(1,22,9,15,28),(3,24,11,17,30),(4,12,19,25,18),(6,14,21,27,8)], [(1,22,9,15,28),(2,23,10,16,29),(3,17,24,30,11),(4,12,19,25,18),(5,13,20,26,7),(6,27,14,8,21)], [(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(30,78);

Polynomial with Galois group C52⋊C12 over ℚ
actionf(x)Disc(f)
15T19x15+x10-2x5-1515·710

Matrix representation of C52⋊C12 in GL12(ℤ)

010000000000
001000000000
000100000000
-1-1-1-100000000
000001000000
000000100000
000000010000
0000-1-1-1-10000
000000000001
00000000-1-1-1-1
000000001000
000000000100
,
100000000000
010000000000
001000000000
000100000000
0000-1-1-1-10000
000010000000
000001000000
000000100000
000000000100
000000000010
000000000001
00000000-1-1-1-1
,
000010000000
000000010000
000001000000
0000-1-1-1-10000
000000001000
000000000001
000000000100
00000000-1-1-1-1
100000000000
000100000000
010000000000
-1-1-1-100000000

G:=sub<GL(12,Integers())| [0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0],[1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,0,0,0,0,-1,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1],[0,0,0,0,0,0,0,0,1,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,-1,1,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0] >;

C52⋊C12 in GAP, Magma, Sage, TeX

C_5^2\rtimes C_{12}
% in TeX

G:=Group("C5^2:C12");
// GroupNames label

G:=SmallGroup(300,24);
// by ID

G=gap.SmallGroup(300,24);
# by ID

G:=PCGroup([5,-2,-3,-2,-5,5,30,483,1928,173,3004,2859,1014]);
// Polycyclic

G:=Group<a,b,c|a^5=b^5=c^12=1,a*b=b*a,c*a*c^-1=a^-1*b,c*b*c^-1=a^3*b^3>;
// generators/relations

Export

Subgroup lattice of C52⋊C12 in TeX
Character table of C52⋊C12 in TeX

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