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## G = C2×C52⋊S3order 300 = 22·3·52

### Direct product of C2 and C52⋊S3

Aliases: C2×C52⋊S3, C522D6, (C5×C10)⋊S3, C52⋊C32C22, (C2×C52⋊C3)⋊1C2, SmallGroup(300,26)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — C52⋊C3 — C2×C52⋊S3
 Chief series C1 — C52 — C52⋊C3 — C52⋊S3 — C2×C52⋊S3
 Lower central C52⋊C3 — C2×C52⋊S3
 Upper central C1 — C2

Generators and relations for C2×C52⋊S3
G = < a,b,c,d,e | a2=b5=c5=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=bc3, be=eb, dcd-1=b-1c3, ece=b-1c-1, ede=d-1 >

15C2
15C2
25C3
3C5
3C5
15C22
25S3
25S3
25C6
3D5
3D5
3C10
3C10
15C10
15C10
25D6
3D10
15C2×C10

Character table of C2×C52⋊S3

 class 1 2A 2B 2C 3 5A 5B 5C 5D 5E 5F 6 10A 10B 10C 10D 10E 10F 10G 10H 10I 10J 10K 10L 10M 10N size 1 1 15 15 50 3 3 3 3 6 6 50 3 3 3 3 6 6 15 15 15 15 15 15 15 15 ρ1 1 1 1 1 1 1 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 -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 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 -1 -1 1 1 1 1 linear of order 2 ρ5 2 2 0 0 -1 2 2 2 2 2 2 -1 2 2 2 2 2 2 0 0 0 0 0 0 0 0 orthogonal lifted from S3 ρ6 2 -2 0 0 -1 2 2 2 2 2 2 1 -2 -2 -2 -2 -2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D6 ρ7 3 -3 -1 1 0 2ζ54+ζ52 ζ53+2ζ5 2ζ52+ζ5 ζ54+2ζ53 1-√5/2 1+√5/2 0 -2ζ54-ζ52 -2ζ52-ζ5 -ζ54-2ζ53 -ζ53-2ζ5 -1-√5/2 -1+√5/2 ζ52 ζ53 ζ54 ζ5 -ζ5 -ζ52 -ζ53 -ζ54 complex faithful ρ8 3 3 -1 -1 0 ζ54+2ζ53 2ζ52+ζ5 2ζ54+ζ52 ζ53+2ζ5 1+√5/2 1-√5/2 0 ζ54+2ζ53 2ζ54+ζ52 ζ53+2ζ5 2ζ52+ζ5 1-√5/2 1+√5/2 -ζ54 -ζ5 -ζ53 -ζ52 -ζ52 -ζ54 -ζ5 -ζ53 complex lifted from C52⋊S3 ρ9 3 3 1 1 0 2ζ52+ζ5 ζ54+2ζ53 ζ53+2ζ5 2ζ54+ζ52 1+√5/2 1-√5/2 0 2ζ52+ζ5 ζ53+2ζ5 2ζ54+ζ52 ζ54+2ζ53 1-√5/2 1+√5/2 ζ5 ζ54 ζ52 ζ53 ζ53 ζ5 ζ54 ζ52 complex lifted from C52⋊S3 ρ10 3 -3 1 -1 0 2ζ52+ζ5 ζ54+2ζ53 ζ53+2ζ5 2ζ54+ζ52 1+√5/2 1-√5/2 0 -2ζ52-ζ5 -ζ53-2ζ5 -2ζ54-ζ52 -ζ54-2ζ53 -1+√5/2 -1-√5/2 -ζ5 -ζ54 -ζ52 -ζ53 ζ53 ζ5 ζ54 ζ52 complex faithful ρ11 3 -3 1 -1 0 ζ54+2ζ53 2ζ52+ζ5 2ζ54+ζ52 ζ53+2ζ5 1+√5/2 1-√5/2 0 -ζ54-2ζ53 -2ζ54-ζ52 -ζ53-2ζ5 -2ζ52-ζ5 -1+√5/2 -1-√5/2 -ζ54 -ζ5 -ζ53 -ζ52 ζ52 ζ54 ζ5 ζ53 complex faithful ρ12 3 3 -1 -1 0 2ζ54+ζ52 ζ53+2ζ5 2ζ52+ζ5 ζ54+2ζ53 1-√5/2 1+√5/2 0 2ζ54+ζ52 2ζ52+ζ5 ζ54+2ζ53 ζ53+2ζ5 1+√5/2 1-√5/2 -ζ52 -ζ53 -ζ54 -ζ5 -ζ5 -ζ52 -ζ53 -ζ54 complex lifted from C52⋊S3 ρ13 3 3 -1 -1 0 2ζ52+ζ5 ζ54+2ζ53 ζ53+2ζ5 2ζ54+ζ52 1+√5/2 1-√5/2 0 2ζ52+ζ5 ζ53+2ζ5 2ζ54+ζ52 ζ54+2ζ53 1-√5/2 1+√5/2 -ζ5 -ζ54 -ζ52 -ζ53 -ζ53 -ζ5 -ζ54 -ζ52 complex lifted from C52⋊S3 ρ14 3 -3 -1 1 0 2ζ52+ζ5 ζ54+2ζ53 ζ53+2ζ5 2ζ54+ζ52 1+√5/2 1-√5/2 0 -2ζ52-ζ5 -ζ53-2ζ5 -2ζ54-ζ52 -ζ54-2ζ53 -1+√5/2 -1-√5/2 ζ5 ζ54 ζ52 ζ53 -ζ53 -ζ5 -ζ54 -ζ52 complex faithful ρ15 3 -3 1 -1 0 ζ53+2ζ5 2ζ54+ζ52 ζ54+2ζ53 2ζ52+ζ5 1-√5/2 1+√5/2 0 -ζ53-2ζ5 -ζ54-2ζ53 -2ζ52-ζ5 -2ζ54-ζ52 -1-√5/2 -1+√5/2 -ζ53 -ζ52 -ζ5 -ζ54 ζ54 ζ53 ζ52 ζ5 complex faithful ρ16 3 3 1 1 0 ζ53+2ζ5 2ζ54+ζ52 ζ54+2ζ53 2ζ52+ζ5 1-√5/2 1+√5/2 0 ζ53+2ζ5 ζ54+2ζ53 2ζ52+ζ5 2ζ54+ζ52 1+√5/2 1-√5/2 ζ53 ζ52 ζ5 ζ54 ζ54 ζ53 ζ52 ζ5 complex lifted from C52⋊S3 ρ17 3 -3 -1 1 0 ζ53+2ζ5 2ζ54+ζ52 ζ54+2ζ53 2ζ52+ζ5 1-√5/2 1+√5/2 0 -ζ53-2ζ5 -ζ54-2ζ53 -2ζ52-ζ5 -2ζ54-ζ52 -1-√5/2 -1+√5/2 ζ53 ζ52 ζ5 ζ54 -ζ54 -ζ53 -ζ52 -ζ5 complex faithful ρ18 3 3 -1 -1 0 ζ53+2ζ5 2ζ54+ζ52 ζ54+2ζ53 2ζ52+ζ5 1-√5/2 1+√5/2 0 ζ53+2ζ5 ζ54+2ζ53 2ζ52+ζ5 2ζ54+ζ52 1+√5/2 1-√5/2 -ζ53 -ζ52 -ζ5 -ζ54 -ζ54 -ζ53 -ζ52 -ζ5 complex lifted from C52⋊S3 ρ19 3 -3 1 -1 0 2ζ54+ζ52 ζ53+2ζ5 2ζ52+ζ5 ζ54+2ζ53 1-√5/2 1+√5/2 0 -2ζ54-ζ52 -2ζ52-ζ5 -ζ54-2ζ53 -ζ53-2ζ5 -1-√5/2 -1+√5/2 -ζ52 -ζ53 -ζ54 -ζ5 ζ5 ζ52 ζ53 ζ54 complex faithful ρ20 3 3 1 1 0 ζ54+2ζ53 2ζ52+ζ5 2ζ54+ζ52 ζ53+2ζ5 1+√5/2 1-√5/2 0 ζ54+2ζ53 2ζ54+ζ52 ζ53+2ζ5 2ζ52+ζ5 1-√5/2 1+√5/2 ζ54 ζ5 ζ53 ζ52 ζ52 ζ54 ζ5 ζ53 complex lifted from C52⋊S3 ρ21 3 -3 -1 1 0 ζ54+2ζ53 2ζ52+ζ5 2ζ54+ζ52 ζ53+2ζ5 1+√5/2 1-√5/2 0 -ζ54-2ζ53 -2ζ54-ζ52 -ζ53-2ζ5 -2ζ52-ζ5 -1+√5/2 -1-√5/2 ζ54 ζ5 ζ53 ζ52 -ζ52 -ζ54 -ζ5 -ζ53 complex faithful ρ22 3 3 1 1 0 2ζ54+ζ52 ζ53+2ζ5 2ζ52+ζ5 ζ54+2ζ53 1-√5/2 1+√5/2 0 2ζ54+ζ52 2ζ52+ζ5 ζ54+2ζ53 ζ53+2ζ5 1+√5/2 1-√5/2 ζ52 ζ53 ζ54 ζ5 ζ5 ζ52 ζ53 ζ54 complex lifted from C52⋊S3 ρ23 6 6 0 0 0 1+√5 1+√5 1-√5 1-√5 -3+√5/2 -3-√5/2 0 1+√5 1-√5 1-√5 1+√5 -3-√5/2 -3+√5/2 0 0 0 0 0 0 0 0 orthogonal lifted from C52⋊S3 ρ24 6 -6 0 0 0 1+√5 1+√5 1-√5 1-√5 -3+√5/2 -3-√5/2 0 -1-√5 -1+√5 -1+√5 -1-√5 3+√5/2 3-√5/2 0 0 0 0 0 0 0 0 orthogonal faithful ρ25 6 -6 0 0 0 1-√5 1-√5 1+√5 1+√5 -3-√5/2 -3+√5/2 0 -1+√5 -1-√5 -1-√5 -1+√5 3-√5/2 3+√5/2 0 0 0 0 0 0 0 0 orthogonal faithful ρ26 6 6 0 0 0 1-√5 1-√5 1+√5 1+√5 -3-√5/2 -3+√5/2 0 1-√5 1+√5 1+√5 1-√5 -3+√5/2 -3-√5/2 0 0 0 0 0 0 0 0 orthogonal lifted from C52⋊S3

Permutation representations of C2×C52⋊S3
On 30 points - transitive group 30T77
Generators in S30
(1 7)(2 6)(3 9)(4 8)(5 10)(11 30)(12 26)(13 27)(14 28)(15 29)(16 24)(17 25)(18 21)(19 22)(20 23)
(11 12 13 14 15)(16 17 18 19 20)(21 22 23 24 25)(26 27 28 29 30)
(1 5 4 3 2)(6 7 10 8 9)(11 14 12 15 13)(16 17 18 19 20)(21 22 23 24 25)(26 29 27 30 28)
(1 20 27)(2 18 29)(3 16 26)(4 19 28)(5 17 30)(6 21 15)(7 23 13)(8 22 14)(9 24 12)(10 25 11)
(1 7)(2 10)(3 8)(4 9)(5 6)(11 18)(12 19)(13 20)(14 16)(15 17)(21 30)(22 26)(23 27)(24 28)(25 29)

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

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

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

G:=TransitiveGroup(30,77);

On 30 points - transitive group 30T81
Generators in S30
(1 17)(2 18)(3 19)(4 20)(5 16)(6 21)(7 22)(8 23)(9 24)(10 25)(11 26)(12 27)(13 28)(14 29)(15 30)
(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 3 5 2 4)(6 10 9 8 7)(16 18 20 17 19)(21 25 24 23 22)
(1 11 6)(2 14 9)(3 12 7)(4 15 10)(5 13 8)(16 28 23)(17 26 21)(18 29 24)(19 27 22)(20 30 25)
(1 17)(2 18)(3 19)(4 20)(5 16)(6 26)(7 27)(8 28)(9 29)(10 30)(11 21)(12 22)(13 23)(14 24)(15 25)

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

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

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

G:=TransitiveGroup(30,81);

Matrix representation of C2×C52⋊S3 in GL3(𝔽11) generated by

 10 0 0 0 10 0 0 0 10
,
 1 3 7 9 7 10 8 8 2
,
 1 5 7 2 4 9 1 0 10
,
 6 2 8 5 10 7 5 1 6
,
 1 4 1 0 1 0 0 3 10
G:=sub<GL(3,GF(11))| [10,0,0,0,10,0,0,0,10],[1,9,8,3,7,8,7,10,2],[1,2,1,5,4,0,7,9,10],[6,5,5,2,10,1,8,7,6],[1,0,0,4,1,3,1,0,10] >;

C2×C52⋊S3 in GAP, Magma, Sage, TeX

C_2\times C_5^2\rtimes S_3
% in TeX

G:=Group("C2xC5^2:S3");
// GroupNames label

G:=SmallGroup(300,26);
// by ID

G=gap.SmallGroup(300,26);
# by ID

G:=PCGroup([5,-2,-2,-3,-5,5,122,973,7204,1439]);
// Polycyclic

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

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