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## G = C52⋊Dic3order 300 = 22·3·52

### The semidirect product of C52 and Dic3 acting faithfully

Aliases: C52⋊Dic3, C5⋊D5.S3, C52⋊C31C4, C52⋊C6.2C2, SmallGroup(300,23)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — C52⋊C3 — C52⋊Dic3
 Chief series C1 — C52 — C52⋊C3 — C52⋊C6 — C52⋊Dic3
 Lower central C52⋊C3 — C52⋊Dic3
 Upper central C1

Generators and relations for C52⋊Dic3
G = < a,b,c,d | a5=b5=c6=1, d2=c3, ab=ba, cac-1=a-1b2, dad-1=a2b, cbc-1=ab2, dbd-1=b3, dcd-1=c-1 >

25C2
25C3
3C5
3C5
75C4
25C6
15D5
15D5
25Dic3
15F5
15F5

Character table of C52⋊Dic3

 class 1 2 3 4A 4B 5A 5B 6 size 1 25 50 75 75 12 12 50 ρ1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 -1 -1 1 1 1 linear of order 2 ρ3 1 -1 1 i -i 1 1 -1 linear of order 4 ρ4 1 -1 1 -i i 1 1 -1 linear of order 4 ρ5 2 2 -1 0 0 2 2 -1 orthogonal lifted from S3 ρ6 2 -2 -1 0 0 2 2 1 symplectic lifted from Dic3, Schur index 2 ρ7 12 0 0 0 0 2 -3 0 orthogonal faithful ρ8 12 0 0 0 0 -3 2 0 orthogonal faithful

Permutation representations of C52⋊Dic3
On 15 points - transitive group 15T17
Generators in S15
(1 8 13 10 5)(2 9 14 11 6)
(1 8 13 10 5)(2 11 9 6 14)(3 15 7 4 12)
(1 2 3)(4 5 6 7 8 9)(10 11 12 13 14 15)
(2 3)(4 11 7 14)(5 10 8 13)(6 15 9 12)

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

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

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

G:=TransitiveGroup(15,17);

On 25 points: primitive - transitive group 25T28
Generators in S25
(1 21 13 10 24)(2 6 9 16 8)(3 5 11 19 12)(4 17 23 22 15)(7 18 25 20 14)
(1 2 14 17 5)(3 24 8 20 4)(6 7 23 11 21)(9 18 22 19 13)(10 16 25 15 12)
(2 3 4 5 6 7)(8 9 10 11 12 13)(14 15 16 17 18 19)(20 21 22 23 24 25)
(2 14 5 17)(3 19 6 16)(4 18 7 15)(8 22 11 25)(9 21 12 24)(10 20 13 23)

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

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

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

G:=TransitiveGroup(25,28);

On 30 points - transitive group 30T71
Generators in S30
(2 19 25 28 22)(3 20 26 29 23)(4 11 18 15 8)(6 10 17 14 7)
(1 27 24 21 30)(2 19 25 28 22)(3 29 20 23 26)(4 15 11 8 18)(5 13 9 12 16)(6 10 17 14 7)
(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)
(1 4)(2 6)(3 5)(7 28 10 25)(8 27 11 30)(9 26 12 29)(13 20 16 23)(14 19 17 22)(15 24 18 21)

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

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

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

G:=TransitiveGroup(30,71);

Polynomial with Galois group C52⋊Dic3 over ℚ
actionf(x)Disc(f)
15T17x15+45x13+630x11+162x10+3625x9+1350x8+9000x7+1800x6+7305x5-1375x3-2250x-450222·322·540·116·134·1272·1949245823572

Matrix representation of C52⋊Dic3 in GL12(ℤ)

 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 -1 -1 -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 -1 -1 -1 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 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 -1 -1 -1 -1
,
 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 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 -1 -1 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 -1 -1 -1 -1 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 1 0 0 0 0 0 0 0 0 0 0 0 -1 -1 -1 -1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 -1 -1 -1 -1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 -1 -1 -1 -1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0
,
 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 -1 -1 -1 -1 0 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 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 -1 -1 -1 -1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 -1 -1 -1 -1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0

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,-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,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-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],[1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-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,-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],[0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,0,0,0,0,-1,1,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0],[1,0,-1,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0] >;

C52⋊Dic3 in GAP, Magma, Sage, TeX

C_5^2\rtimes {\rm Dic}_3
% in TeX

G:=Group("C5^2:Dic3");
// GroupNames label

G:=SmallGroup(300,23);
// by ID

G=gap.SmallGroup(300,23);
# by ID

G:=PCGroup([5,-2,-2,-3,-5,5,10,122,5523,488,793,3004,3009,464]);
// Polycyclic

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

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