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## G = C2×C52⋊C6order 300 = 22·3·52

### Direct product of C2 and C52⋊C6

Aliases: C2×C52⋊C6, C5⋊D5⋊C6, (C5×C10)⋊C6, C52⋊(C2×C6), C52⋊C33C22, (C2×C5⋊D5)⋊C3, (C2×C52⋊C3)⋊2C2, SmallGroup(300,27)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — C2×C52⋊C6
 Chief series C1 — C52 — C52⋊C3 — C52⋊C6 — C2×C52⋊C6
 Lower central C52 — C2×C52⋊C6
 Upper central C1 — C2

Generators and relations for C2×C52⋊C6
G = < a,b,c,d | a2=b5=c5=d6=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b2c3, dcd-1=b-1c-1 >

25C2
25C2
25C3
3C5
3C5
25C22
25C6
25C6
25C6
3C10
3C10
15D5
15D5
15D5
15D5
25C2×C6
15D10
15D10

Character table of C2×C52⋊C6

 class 1 2A 2B 2C 3A 3B 5A 5B 5C 5D 6A 6B 6C 6D 6E 6F 10A 10B 10C 10D size 1 1 25 25 25 25 6 6 6 6 25 25 25 25 25 25 6 6 6 6 ρ1 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 1 1 1 linear of order 2 ρ3 1 -1 -1 1 1 1 1 1 1 1 -1 1 1 -1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ4 1 -1 1 -1 1 1 1 1 1 1 1 -1 -1 -1 1 -1 -1 -1 -1 -1 linear of order 2 ρ5 1 1 1 1 ζ3 ζ32 1 1 1 1 ζ3 ζ32 ζ3 ζ3 ζ32 ζ32 1 1 1 1 linear of order 3 ρ6 1 1 -1 -1 ζ32 ζ3 1 1 1 1 ζ6 ζ65 ζ6 ζ32 ζ65 ζ3 1 1 1 1 linear of order 6 ρ7 1 1 1 1 ζ32 ζ3 1 1 1 1 ζ32 ζ3 ζ32 ζ32 ζ3 ζ3 1 1 1 1 linear of order 3 ρ8 1 -1 -1 1 ζ3 ζ32 1 1 1 1 ζ65 ζ32 ζ3 ζ65 ζ6 ζ6 -1 -1 -1 -1 linear of order 6 ρ9 1 -1 1 -1 ζ3 ζ32 1 1 1 1 ζ3 ζ6 ζ65 ζ65 ζ32 ζ6 -1 -1 -1 -1 linear of order 6 ρ10 1 -1 -1 1 ζ32 ζ3 1 1 1 1 ζ6 ζ3 ζ32 ζ6 ζ65 ζ65 -1 -1 -1 -1 linear of order 6 ρ11 1 -1 1 -1 ζ32 ζ3 1 1 1 1 ζ32 ζ65 ζ6 ζ6 ζ3 ζ65 -1 -1 -1 -1 linear of order 6 ρ12 1 1 -1 -1 ζ3 ζ32 1 1 1 1 ζ65 ζ6 ζ65 ζ3 ζ6 ζ32 1 1 1 1 linear of order 6 ρ13 6 -6 0 0 0 0 1+√5 -3-√5/2 -3+√5/2 1-√5 0 0 0 0 0 0 3+√5/2 3-√5/2 -1+√5 -1-√5 orthogonal faithful ρ14 6 6 0 0 0 0 1+√5 -3-√5/2 -3+√5/2 1-√5 0 0 0 0 0 0 -3-√5/2 -3+√5/2 1-√5 1+√5 orthogonal lifted from C52⋊C6 ρ15 6 6 0 0 0 0 1-√5 -3+√5/2 -3-√5/2 1+√5 0 0 0 0 0 0 -3+√5/2 -3-√5/2 1+√5 1-√5 orthogonal lifted from C52⋊C6 ρ16 6 6 0 0 0 0 -3-√5/2 1-√5 1+√5 -3+√5/2 0 0 0 0 0 0 1-√5 1+√5 -3+√5/2 -3-√5/2 orthogonal lifted from C52⋊C6 ρ17 6 -6 0 0 0 0 1-√5 -3+√5/2 -3-√5/2 1+√5 0 0 0 0 0 0 3-√5/2 3+√5/2 -1-√5 -1+√5 orthogonal faithful ρ18 6 -6 0 0 0 0 -3-√5/2 1-√5 1+√5 -3+√5/2 0 0 0 0 0 0 -1+√5 -1-√5 3-√5/2 3+√5/2 orthogonal faithful ρ19 6 -6 0 0 0 0 -3+√5/2 1+√5 1-√5 -3-√5/2 0 0 0 0 0 0 -1-√5 -1+√5 3+√5/2 3-√5/2 orthogonal faithful ρ20 6 6 0 0 0 0 -3+√5/2 1+√5 1-√5 -3-√5/2 0 0 0 0 0 0 1+√5 1-√5 -3-√5/2 -3+√5/2 orthogonal lifted from C52⋊C6

Permutation representations of C2×C52⋊C6
On 30 points - transitive group 30T68
Generators in S30
(1 6)(2 4)(3 5)(7 25)(8 26)(9 27)(10 28)(11 29)(12 30)(13 23)(14 24)(15 19)(16 20)(17 21)(18 22)
(1 9 14 17 12)(2 18 10 7 15)(3 16 8 11 13)(4 22 28 25 19)(5 20 26 29 23)(6 27 24 21 30)
(1 14 12 9 17)(2 15 7 10 18)(4 19 25 28 22)(6 24 30 27 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)| (1,6)(2,4)(3,5)(7,25)(8,26)(9,27)(10,28)(11,29)(12,30)(13,23)(14,24)(15,19)(16,20)(17,21)(18,22), (1,9,14,17,12)(2,18,10,7,15)(3,16,8,11,13)(4,22,28,25,19)(5,20,26,29,23)(6,27,24,21,30), (1,14,12,9,17)(2,15,7,10,18)(4,19,25,28,22)(6,24,30,27,21), (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( (1,6)(2,4)(3,5)(7,25)(8,26)(9,27)(10,28)(11,29)(12,30)(13,23)(14,24)(15,19)(16,20)(17,21)(18,22), (1,9,14,17,12)(2,18,10,7,15)(3,16,8,11,13)(4,22,28,25,19)(5,20,26,29,23)(6,27,24,21,30), (1,14,12,9,17)(2,15,7,10,18)(4,19,25,28,22)(6,24,30,27,21), (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([[(1,6),(2,4),(3,5),(7,25),(8,26),(9,27),(10,28),(11,29),(12,30),(13,23),(14,24),(15,19),(16,20),(17,21),(18,22)], [(1,9,14,17,12),(2,18,10,7,15),(3,16,8,11,13),(4,22,28,25,19),(5,20,26,29,23),(6,27,24,21,30)], [(1,14,12,9,17),(2,15,7,10,18),(4,19,25,28,22),(6,24,30,27,21)], [(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(30,68);

Matrix representation of C2×C52⋊C6 in GL6(𝔽31)

 30 0 0 0 0 0 0 30 0 0 0 0 0 0 30 0 0 0 0 0 0 30 0 0 0 0 0 0 30 0 0 0 0 0 0 30
,
 19 30 0 0 0 0 20 30 0 0 0 0 0 0 12 18 0 0 0 0 12 0 0 0 0 0 0 0 19 30 0 0 0 0 20 30
,
 30 1 0 0 0 0 11 19 0 0 0 0 0 0 19 30 0 0 0 0 20 30 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 0 19 30 0 0 0 0 19 12 0 0 0 0 0 0 19 30 0 0 0 0 19 12 19 30 0 0 0 0 19 12 0 0 0 0

G:=sub<GL(6,GF(31))| [30,0,0,0,0,0,0,30,0,0,0,0,0,0,30,0,0,0,0,0,0,30,0,0,0,0,0,0,30,0,0,0,0,0,0,30],[19,20,0,0,0,0,30,30,0,0,0,0,0,0,12,12,0,0,0,0,18,0,0,0,0,0,0,0,19,20,0,0,0,0,30,30],[30,11,0,0,0,0,1,19,0,0,0,0,0,0,19,20,0,0,0,0,30,30,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,0,0,0,19,19,0,0,0,0,30,12,19,19,0,0,0,0,30,12,0,0,0,0,0,0,19,19,0,0,0,0,30,12,0,0] >;

C2×C52⋊C6 in GAP, Magma, Sage, TeX

C_2\times C_5^2\rtimes C_6
% in TeX

G:=Group("C2xC5^2:C6");
// GroupNames label

G:=SmallGroup(300,27);
// by ID

G=gap.SmallGroup(300,27);
# by ID

G:=PCGroup([5,-2,-2,-3,-5,5,963,793,6004,464]);
// Polycyclic

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

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