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G = C52⋊Dic3order 300 = 22·3·52

The semidirect product of C52 and Dic3 acting faithfully

non-abelian, soluble, monomial, A-group

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

Series: Derived Chief Lower central Upper central

C1C52C52⋊C3 — C52⋊Dic3
C1C52C52⋊C3C52⋊C6 — C52⋊Dic3
C52⋊C3 — C52⋊Dic3
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
3C52⋊C4

Character table of C52⋊Dic3

 class 1234A4B5A5B6
 size 125507575121250
ρ111111111    trivial
ρ2111-1-1111    linear of order 2
ρ31-11i-i11-1    linear of order 4
ρ41-11-ii11-1    linear of order 4
ρ522-10022-1    orthogonal lifted from S3
ρ62-2-100221    symplectic lifted from Dic3, Schur index 2
ρ71200002-30    orthogonal faithful
ρ8120000-320    orthogonal faithful

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

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

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

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

G:=TransitiveGroup(15,17);

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

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

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

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

G:=TransitiveGroup(25,28);

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

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

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

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

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)

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

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

Export

Subgroup lattice of C52⋊Dic3 in TeX
Character table of C52⋊Dic3 in TeX

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