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G = C5×C3⋊S3order 90 = 2·32·5

Direct product of C5 and C3⋊S3

direct product, metabelian, supersoluble, monomial, A-group

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

Series: Derived Chief Lower central Upper central

C1C32 — C5×C3⋊S3
C1C3C32C3×C15 — C5×C3⋊S3
C32 — C5×C3⋊S3
C1C5

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 >

9C2
3S3
3S3
3S3
3S3
9C10
3C5×S3
3C5×S3
3C5×S3
3C5×S3

Character table of C5×C3⋊S3

 class 123A3B3C3D5A5B5C5D10A10B10C10D15A15B15C15D15E15F15G15H15I15J15K15L15M15N15O15P
 size 192222111199992222222222222222
ρ1111111111111111111111111111111    trivial
ρ21-111111111-1-1-1-11111111111111111    linear of order 2
ρ3111111ζ53ζ5ζ54ζ52ζ54ζ5ζ52ζ53ζ5ζ5ζ53ζ53ζ53ζ53ζ5ζ54ζ54ζ54ζ54ζ5ζ52ζ52ζ52ζ52    linear of order 5
ρ41-11111ζ54ζ53ζ52ζ55253554ζ53ζ53ζ54ζ54ζ54ζ54ζ53ζ52ζ52ζ52ζ52ζ53ζ5ζ5ζ5ζ5    linear of order 10
ρ51-11111ζ5ζ52ζ53ζ545352545ζ52ζ52ζ5ζ5ζ5ζ5ζ52ζ53ζ53ζ53ζ53ζ52ζ54ζ54ζ54ζ54    linear of order 10
ρ61-11111ζ52ζ54ζ5ζ535545352ζ54ζ54ζ52ζ52ζ52ζ52ζ54ζ5ζ5ζ5ζ5ζ54ζ53ζ53ζ53ζ53    linear of order 10
ρ7111111ζ5ζ52ζ53ζ54ζ53ζ52ζ54ζ5ζ52ζ52ζ5ζ5ζ5ζ5ζ52ζ53ζ53ζ53ζ53ζ52ζ54ζ54ζ54ζ54    linear of order 5
ρ8111111ζ54ζ53ζ52ζ5ζ52ζ53ζ5ζ54ζ53ζ53ζ54ζ54ζ54ζ54ζ53ζ52ζ52ζ52ζ52ζ53ζ5ζ5ζ5ζ5    linear of order 5
ρ9111111ζ52ζ54ζ5ζ53ζ5ζ54ζ53ζ52ζ54ζ54ζ52ζ52ζ52ζ52ζ54ζ5ζ5ζ5ζ5ζ54ζ53ζ53ζ53ζ53    linear of order 5
ρ101-11111ζ53ζ5ζ54ζ525455253ζ5ζ5ζ53ζ53ζ53ζ53ζ5ζ54ζ54ζ54ζ54ζ5ζ52ζ52ζ52ζ52    linear of order 10
ρ11202-1-1-122220000-12-12-1-1-1-12-1-1-1-12-1-1    orthogonal lifted from S3
ρ1220-1-12-122220000-1-1-1-1-12-1-1-1-122-1-1-12    orthogonal lifted from S3
ρ1320-1-1-1222220000-1-12-1-1-122-1-1-1-12-1-1-1    orthogonal lifted from S3
ρ1420-12-1-1222200002-1-1-12-1-1-1-12-1-1-1-12-1    orthogonal lifted from S3
ρ1520-1-12-1535545200005553535353554545454552525252    complex lifted from C5×S3
ρ1620-1-12-1545352500005353545454545352525252535555    complex lifted from C5×S3
ρ17202-1-1-1552535400005252555552535353535254545454    complex lifted from C5×S3
ρ1820-1-1-12552535400005252555552535353535254545454    complex lifted from C5×S3
ρ1920-12-1-1535545200005553535353554545454552525252    complex lifted from C5×S3
ρ2020-12-1-1552535400005252555552535353535254545454    complex lifted from C5×S3
ρ2120-1-12-1552535400005252555552535353535254545454    complex lifted from C5×S3
ρ2220-1-1-12535545200005553535353554545454552525252    complex lifted from C5×S3
ρ23202-1-1-1545352500005353545454545352525252535555    complex lifted from C5×S3
ρ2420-1-1-12545352500005353545454545352525252535555    complex lifted from C5×S3
ρ2520-12-1-1545352500005353545454545352525252535555    complex lifted from C5×S3
ρ2620-1-1-12525455300005454525252525455555453535353    complex lifted from C5×S3
ρ2720-12-1-1525455300005454525252525455555453535353    complex lifted from C5×S3
ρ28202-1-1-1525455300005454525252525455555453535353    complex lifted from C5×S3
ρ29202-1-1-1535545200005553535353554545454552525252    complex lifted from C5×S3
ρ3020-1-12-1525455300005454525252525455555453535353    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 26 22)(12 27 23)(13 28 24)(14 29 25)(15 30 21)(16 41 40)(17 42 36)(18 43 37)(19 44 38)(20 45 39)
(1 27 42)(2 28 43)(3 29 44)(4 30 45)(5 26 41)(6 21 39)(7 22 40)(8 23 36)(9 24 37)(10 25 38)(11 16 32)(12 17 33)(13 18 34)(14 19 35)(15 20 31)
(6 31)(7 32)(8 33)(9 34)(10 35)(11 40)(12 36)(13 37)(14 38)(15 39)(16 22)(17 23)(18 24)(19 25)(20 21)(26 41)(27 42)(28 43)(29 44)(30 45)

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

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

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

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

8000
0800
0020
0002
,
1000
0100
00301
00300
,
03000
13000
00030
00130
,
13000
03000
0001
0010
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

Subgroup lattice of C5×C3⋊S3 in TeX
Character table of C5×C3⋊S3 in TeX

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