Copied to
clipboard

## G = C5×C3⋊S3order 90 = 2·32·5

### Direct product of C5 and C3⋊S3

Aliases: C5×C3⋊S3, C153S3, C322C10, C3⋊(C5×S3), (C3×C15)⋊5C2, SmallGroup(90,8)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C5×C3⋊S3
 Chief series C1 — C3 — C32 — C3×C15 — C5×C3⋊S3
 Lower central C32 — C5×C3⋊S3
 Upper central C1 — C5

Generators and relations for C5×C3⋊S3
G = < a,b,c,d | a5=b3=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >

Character table of C5×C3⋊S3

 class 1 2 3A 3B 3C 3D 5A 5B 5C 5D 10A 10B 10C 10D 15A 15B 15C 15D 15E 15F 15G 15H 15I 15J 15K 15L 15M 15N 15O 15P size 1 9 2 2 2 2 1 1 1 1 9 9 9 9 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 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 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 1 1 1 1 linear of order 2 ρ3 1 1 1 1 1 1 ζ53 ζ5 ζ54 ζ52 ζ54 ζ5 ζ52 ζ53 ζ5 ζ5 ζ53 ζ53 ζ53 ζ53 ζ5 ζ54 ζ54 ζ54 ζ54 ζ5 ζ52 ζ52 ζ52 ζ52 linear of order 5 ρ4 1 -1 1 1 1 1 ζ54 ζ53 ζ52 ζ5 -ζ52 -ζ53 -ζ5 -ζ54 ζ53 ζ53 ζ54 ζ54 ζ54 ζ54 ζ53 ζ52 ζ52 ζ52 ζ52 ζ53 ζ5 ζ5 ζ5 ζ5 linear of order 10 ρ5 1 -1 1 1 1 1 ζ5 ζ52 ζ53 ζ54 -ζ53 -ζ52 -ζ54 -ζ5 ζ52 ζ52 ζ5 ζ5 ζ5 ζ5 ζ52 ζ53 ζ53 ζ53 ζ53 ζ52 ζ54 ζ54 ζ54 ζ54 linear of order 10 ρ6 1 -1 1 1 1 1 ζ52 ζ54 ζ5 ζ53 -ζ5 -ζ54 -ζ53 -ζ52 ζ54 ζ54 ζ52 ζ52 ζ52 ζ52 ζ54 ζ5 ζ5 ζ5 ζ5 ζ54 ζ53 ζ53 ζ53 ζ53 linear of order 10 ρ7 1 1 1 1 1 1 ζ5 ζ52 ζ53 ζ54 ζ53 ζ52 ζ54 ζ5 ζ52 ζ52 ζ5 ζ5 ζ5 ζ5 ζ52 ζ53 ζ53 ζ53 ζ53 ζ52 ζ54 ζ54 ζ54 ζ54 linear of order 5 ρ8 1 1 1 1 1 1 ζ54 ζ53 ζ52 ζ5 ζ52 ζ53 ζ5 ζ54 ζ53 ζ53 ζ54 ζ54 ζ54 ζ54 ζ53 ζ52 ζ52 ζ52 ζ52 ζ53 ζ5 ζ5 ζ5 ζ5 linear of order 5 ρ9 1 1 1 1 1 1 ζ52 ζ54 ζ5 ζ53 ζ5 ζ54 ζ53 ζ52 ζ54 ζ54 ζ52 ζ52 ζ52 ζ52 ζ54 ζ5 ζ5 ζ5 ζ5 ζ54 ζ53 ζ53 ζ53 ζ53 linear of order 5 ρ10 1 -1 1 1 1 1 ζ53 ζ5 ζ54 ζ52 -ζ54 -ζ5 -ζ52 -ζ53 ζ5 ζ5 ζ53 ζ53 ζ53 ζ53 ζ5 ζ54 ζ54 ζ54 ζ54 ζ5 ζ52 ζ52 ζ52 ζ52 linear of order 10 ρ11 2 0 2 -1 -1 -1 2 2 2 2 0 0 0 0 -1 2 -1 2 -1 -1 -1 -1 2 -1 -1 -1 -1 2 -1 -1 orthogonal lifted from S3 ρ12 2 0 -1 -1 2 -1 2 2 2 2 0 0 0 0 -1 -1 -1 -1 -1 2 -1 -1 -1 -1 2 2 -1 -1 -1 2 orthogonal lifted from S3 ρ13 2 0 -1 -1 -1 2 2 2 2 2 0 0 0 0 -1 -1 2 -1 -1 -1 2 2 -1 -1 -1 -1 2 -1 -1 -1 orthogonal lifted from S3 ρ14 2 0 -1 2 -1 -1 2 2 2 2 0 0 0 0 2 -1 -1 -1 2 -1 -1 -1 -1 2 -1 -1 -1 -1 2 -1 orthogonal lifted from S3 ρ15 2 0 -1 -1 2 -1 2ζ53 2ζ5 2ζ54 2ζ52 0 0 0 0 -ζ5 -ζ5 -ζ53 -ζ53 -ζ53 2ζ53 -ζ5 -ζ54 -ζ54 -ζ54 2ζ54 2ζ5 -ζ52 -ζ52 -ζ52 2ζ52 complex lifted from C5×S3 ρ16 2 0 -1 -1 2 -1 2ζ54 2ζ53 2ζ52 2ζ5 0 0 0 0 -ζ53 -ζ53 -ζ54 -ζ54 -ζ54 2ζ54 -ζ53 -ζ52 -ζ52 -ζ52 2ζ52 2ζ53 -ζ5 -ζ5 -ζ5 2ζ5 complex lifted from C5×S3 ρ17 2 0 2 -1 -1 -1 2ζ5 2ζ52 2ζ53 2ζ54 0 0 0 0 -ζ52 2ζ52 -ζ5 2ζ5 -ζ5 -ζ5 -ζ52 -ζ53 2ζ53 -ζ53 -ζ53 -ζ52 -ζ54 2ζ54 -ζ54 -ζ54 complex lifted from C5×S3 ρ18 2 0 -1 -1 -1 2 2ζ5 2ζ52 2ζ53 2ζ54 0 0 0 0 -ζ52 -ζ52 2ζ5 -ζ5 -ζ5 -ζ5 2ζ52 2ζ53 -ζ53 -ζ53 -ζ53 -ζ52 2ζ54 -ζ54 -ζ54 -ζ54 complex lifted from C5×S3 ρ19 2 0 -1 2 -1 -1 2ζ53 2ζ5 2ζ54 2ζ52 0 0 0 0 2ζ5 -ζ5 -ζ53 -ζ53 2ζ53 -ζ53 -ζ5 -ζ54 -ζ54 2ζ54 -ζ54 -ζ5 -ζ52 -ζ52 2ζ52 -ζ52 complex lifted from C5×S3 ρ20 2 0 -1 2 -1 -1 2ζ5 2ζ52 2ζ53 2ζ54 0 0 0 0 2ζ52 -ζ52 -ζ5 -ζ5 2ζ5 -ζ5 -ζ52 -ζ53 -ζ53 2ζ53 -ζ53 -ζ52 -ζ54 -ζ54 2ζ54 -ζ54 complex lifted from C5×S3 ρ21 2 0 -1 -1 2 -1 2ζ5 2ζ52 2ζ53 2ζ54 0 0 0 0 -ζ52 -ζ52 -ζ5 -ζ5 -ζ5 2ζ5 -ζ52 -ζ53 -ζ53 -ζ53 2ζ53 2ζ52 -ζ54 -ζ54 -ζ54 2ζ54 complex lifted from C5×S3 ρ22 2 0 -1 -1 -1 2 2ζ53 2ζ5 2ζ54 2ζ52 0 0 0 0 -ζ5 -ζ5 2ζ53 -ζ53 -ζ53 -ζ53 2ζ5 2ζ54 -ζ54 -ζ54 -ζ54 -ζ5 2ζ52 -ζ52 -ζ52 -ζ52 complex lifted from C5×S3 ρ23 2 0 2 -1 -1 -1 2ζ54 2ζ53 2ζ52 2ζ5 0 0 0 0 -ζ53 2ζ53 -ζ54 2ζ54 -ζ54 -ζ54 -ζ53 -ζ52 2ζ52 -ζ52 -ζ52 -ζ53 -ζ5 2ζ5 -ζ5 -ζ5 complex lifted from C5×S3 ρ24 2 0 -1 -1 -1 2 2ζ54 2ζ53 2ζ52 2ζ5 0 0 0 0 -ζ53 -ζ53 2ζ54 -ζ54 -ζ54 -ζ54 2ζ53 2ζ52 -ζ52 -ζ52 -ζ52 -ζ53 2ζ5 -ζ5 -ζ5 -ζ5 complex lifted from C5×S3 ρ25 2 0 -1 2 -1 -1 2ζ54 2ζ53 2ζ52 2ζ5 0 0 0 0 2ζ53 -ζ53 -ζ54 -ζ54 2ζ54 -ζ54 -ζ53 -ζ52 -ζ52 2ζ52 -ζ52 -ζ53 -ζ5 -ζ5 2ζ5 -ζ5 complex lifted from C5×S3 ρ26 2 0 -1 -1 -1 2 2ζ52 2ζ54 2ζ5 2ζ53 0 0 0 0 -ζ54 -ζ54 2ζ52 -ζ52 -ζ52 -ζ52 2ζ54 2ζ5 -ζ5 -ζ5 -ζ5 -ζ54 2ζ53 -ζ53 -ζ53 -ζ53 complex lifted from C5×S3 ρ27 2 0 -1 2 -1 -1 2ζ52 2ζ54 2ζ5 2ζ53 0 0 0 0 2ζ54 -ζ54 -ζ52 -ζ52 2ζ52 -ζ52 -ζ54 -ζ5 -ζ5 2ζ5 -ζ5 -ζ54 -ζ53 -ζ53 2ζ53 -ζ53 complex lifted from C5×S3 ρ28 2 0 2 -1 -1 -1 2ζ52 2ζ54 2ζ5 2ζ53 0 0 0 0 -ζ54 2ζ54 -ζ52 2ζ52 -ζ52 -ζ52 -ζ54 -ζ5 2ζ5 -ζ5 -ζ5 -ζ54 -ζ53 2ζ53 -ζ53 -ζ53 complex lifted from C5×S3 ρ29 2 0 2 -1 -1 -1 2ζ53 2ζ5 2ζ54 2ζ52 0 0 0 0 -ζ5 2ζ5 -ζ53 2ζ53 -ζ53 -ζ53 -ζ5 -ζ54 2ζ54 -ζ54 -ζ54 -ζ5 -ζ52 2ζ52 -ζ52 -ζ52 complex lifted from C5×S3 ρ30 2 0 -1 -1 2 -1 2ζ52 2ζ54 2ζ5 2ζ53 0 0 0 0 -ζ54 -ζ54 -ζ52 -ζ52 -ζ52 2ζ52 -ζ54 -ζ5 -ζ5 -ζ5 2ζ5 2ζ54 -ζ53 -ζ53 -ζ53 2ζ53 complex lifted from C5×S3

Smallest permutation representation of C5×C3⋊S3
On 45 points
Generators in S45
(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)(31 32 33 34 35)(36 37 38 39 40)(41 42 43 44 45)
(1 8 33)(2 9 34)(3 10 35)(4 6 31)(5 7 32)(11 23 20)(12 24 16)(13 25 17)(14 21 18)(15 22 19)(26 42 36)(27 43 37)(28 44 38)(29 45 39)(30 41 40)
(1 18 42)(2 19 43)(3 20 44)(4 16 45)(5 17 41)(6 12 39)(7 13 40)(8 14 36)(9 15 37)(10 11 38)(21 26 33)(22 27 34)(23 28 35)(24 29 31)(25 30 32)
(6 31)(7 32)(8 33)(9 34)(10 35)(11 28)(12 29)(13 30)(14 26)(15 27)(16 45)(17 41)(18 42)(19 43)(20 44)(21 36)(22 37)(23 38)(24 39)(25 40)

G:=sub<Sym(45)| (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)(31,32,33,34,35)(36,37,38,39,40)(41,42,43,44,45), (1,8,33)(2,9,34)(3,10,35)(4,6,31)(5,7,32)(11,23,20)(12,24,16)(13,25,17)(14,21,18)(15,22,19)(26,42,36)(27,43,37)(28,44,38)(29,45,39)(30,41,40), (1,18,42)(2,19,43)(3,20,44)(4,16,45)(5,17,41)(6,12,39)(7,13,40)(8,14,36)(9,15,37)(10,11,38)(21,26,33)(22,27,34)(23,28,35)(24,29,31)(25,30,32), (6,31)(7,32)(8,33)(9,34)(10,35)(11,28)(12,29)(13,30)(14,26)(15,27)(16,45)(17,41)(18,42)(19,43)(20,44)(21,36)(22,37)(23,38)(24,39)(25,40)>;

G:=Group( (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)(31,32,33,34,35)(36,37,38,39,40)(41,42,43,44,45), (1,8,33)(2,9,34)(3,10,35)(4,6,31)(5,7,32)(11,23,20)(12,24,16)(13,25,17)(14,21,18)(15,22,19)(26,42,36)(27,43,37)(28,44,38)(29,45,39)(30,41,40), (1,18,42)(2,19,43)(3,20,44)(4,16,45)(5,17,41)(6,12,39)(7,13,40)(8,14,36)(9,15,37)(10,11,38)(21,26,33)(22,27,34)(23,28,35)(24,29,31)(25,30,32), (6,31)(7,32)(8,33)(9,34)(10,35)(11,28)(12,29)(13,30)(14,26)(15,27)(16,45)(17,41)(18,42)(19,43)(20,44)(21,36)(22,37)(23,38)(24,39)(25,40) );

G=PermutationGroup([[(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),(31,32,33,34,35),(36,37,38,39,40),(41,42,43,44,45)], [(1,8,33),(2,9,34),(3,10,35),(4,6,31),(5,7,32),(11,23,20),(12,24,16),(13,25,17),(14,21,18),(15,22,19),(26,42,36),(27,43,37),(28,44,38),(29,45,39),(30,41,40)], [(1,18,42),(2,19,43),(3,20,44),(4,16,45),(5,17,41),(6,12,39),(7,13,40),(8,14,36),(9,15,37),(10,11,38),(21,26,33),(22,27,34),(23,28,35),(24,29,31),(25,30,32)], [(6,31),(7,32),(8,33),(9,34),(10,35),(11,28),(12,29),(13,30),(14,26),(15,27),(16,45),(17,41),(18,42),(19,43),(20,44),(21,36),(22,37),(23,38),(24,39),(25,40)]])

C5×C3⋊S3 is a maximal subgroup of   C32⋊Dic5  C5×S32  D15⋊S3

Matrix representation of C5×C3⋊S3 in GL4(𝔽31) generated by

 8 0 0 0 0 8 0 0 0 0 2 0 0 0 0 2
,
 1 0 0 0 0 1 0 0 0 0 30 1 0 0 30 0
,
 0 30 0 0 1 30 0 0 0 0 0 30 0 0 1 30
,
 1 30 0 0 0 30 0 0 0 0 0 1 0 0 1 0
G:=sub<GL(4,GF(31))| [8,0,0,0,0,8,0,0,0,0,2,0,0,0,0,2],[1,0,0,0,0,1,0,0,0,0,30,30,0,0,1,0],[0,1,0,0,30,30,0,0,0,0,0,1,0,0,30,30],[1,0,0,0,30,30,0,0,0,0,0,1,0,0,1,0] >;

C5×C3⋊S3 in GAP, Magma, Sage, TeX

C_5\times C_3\rtimes S_3
% in TeX

G:=Group("C5xC3:S3");
// GroupNames label

G:=SmallGroup(90,8);
// by ID

G=gap.SmallGroup(90,8);
# by ID

G:=PCGroup([4,-2,-5,-3,-3,242,963]);
// Polycyclic

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

Export

׿
×
𝔽