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## G = C3⋊S3×F5order 360 = 23·32·5

### Direct product of C3⋊S3 and F5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C15 — C3⋊S3×F5
 Chief series C1 — C5 — C15 — C3×C15 — C32×D5 — C32×F5 — C3⋊S3×F5
 Lower central C3×C15 — C3⋊S3×F5
 Upper central C1

Generators and relations for C3⋊S3×F5
G = < a,b,c,d,e | a3=b3=c2=d5=e4=1, ab=ba, cac=a-1, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d3 >

Subgroups: 656 in 96 conjugacy classes, 30 normal (14 characteristic)
C1, C2, C3, C4, C22, C5, S3, C6, C2×C4, C32, D5, D5, C10, Dic3, C12, D6, C15, C3⋊S3, C3⋊S3, C3×C6, F5, F5, D10, C4×S3, C5×S3, C3×D5, D15, C3⋊Dic3, C3×C12, C2×C3⋊S3, C2×F5, C3×C15, C3×F5, C3⋊F5, S3×D5, C4×C3⋊S3, C32×D5, C5×C3⋊S3, C3⋊D15, S3×F5, C32×F5, C323F5, D5×C3⋊S3, C3⋊S3×F5
Quotients: C1, C2, C4, C22, S3, C2×C4, D6, C3⋊S3, F5, C4×S3, C2×C3⋊S3, C2×F5, C4×C3⋊S3, S3×F5, C3⋊S3×F5

Character table of C3⋊S3×F5

 class 1 2A 2B 2C 3A 3B 3C 3D 4A 4B 4C 4D 5 6A 6B 6C 6D 10 12A 12B 12C 12D 12E 12F 12G 12H 15A 15B 15C 15D size 1 5 9 45 2 2 2 2 5 5 45 45 4 10 10 10 10 36 10 10 10 10 10 10 10 10 8 8 8 8 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 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 1 1 1 1 linear of order 2 ρ5 1 -1 -1 1 1 1 1 1 i -i -i i 1 -1 -1 -1 -1 -1 i -i -i -i -i i i i 1 1 1 1 linear of order 4 ρ6 1 -1 1 -1 1 1 1 1 i -i i -i 1 -1 -1 -1 -1 1 i -i -i -i -i i i i 1 1 1 1 linear of order 4 ρ7 1 -1 1 -1 1 1 1 1 -i i -i i 1 -1 -1 -1 -1 1 -i i i i i -i -i -i 1 1 1 1 linear of order 4 ρ8 1 -1 -1 1 1 1 1 1 -i i i -i 1 -1 -1 -1 -1 -1 -i i i i i -i -i -i 1 1 1 1 linear of order 4 ρ9 2 2 0 0 2 -1 -1 -1 -2 -2 0 0 2 -1 2 -1 -1 0 1 1 -2 1 1 1 -2 1 2 -1 -1 -1 orthogonal lifted from D6 ρ10 2 2 0 0 -1 -1 -1 2 -2 -2 0 0 2 2 -1 -1 -1 0 1 -2 1 1 1 -2 1 1 -1 -1 2 -1 orthogonal lifted from D6 ρ11 2 2 0 0 -1 2 -1 -1 2 2 0 0 2 -1 -1 2 -1 0 -1 -1 -1 2 -1 -1 -1 2 -1 -1 -1 2 orthogonal lifted from S3 ρ12 2 2 0 0 -1 -1 2 -1 -2 -2 0 0 2 -1 -1 -1 2 0 -2 1 1 1 -2 1 1 1 -1 2 -1 -1 orthogonal lifted from D6 ρ13 2 2 0 0 -1 -1 2 -1 2 2 0 0 2 -1 -1 -1 2 0 2 -1 -1 -1 2 -1 -1 -1 -1 2 -1 -1 orthogonal lifted from S3 ρ14 2 2 0 0 2 -1 -1 -1 2 2 0 0 2 -1 2 -1 -1 0 -1 -1 2 -1 -1 -1 2 -1 2 -1 -1 -1 orthogonal lifted from S3 ρ15 2 2 0 0 -1 2 -1 -1 -2 -2 0 0 2 -1 -1 2 -1 0 1 1 1 -2 1 1 1 -2 -1 -1 -1 2 orthogonal lifted from D6 ρ16 2 2 0 0 -1 -1 -1 2 2 2 0 0 2 2 -1 -1 -1 0 -1 2 -1 -1 -1 2 -1 -1 -1 -1 2 -1 orthogonal lifted from S3 ρ17 2 -2 0 0 -1 -1 2 -1 2i -2i 0 0 2 1 1 1 -2 0 2i i i i -2i -i -i -i -1 2 -1 -1 complex lifted from C4×S3 ρ18 2 -2 0 0 -1 -1 -1 2 -2i 2i 0 0 2 -2 1 1 1 0 i 2i -i -i -i -2i i i -1 -1 2 -1 complex lifted from C4×S3 ρ19 2 -2 0 0 -1 -1 -1 2 2i -2i 0 0 2 -2 1 1 1 0 -i -2i i i i 2i -i -i -1 -1 2 -1 complex lifted from C4×S3 ρ20 2 -2 0 0 -1 2 -1 -1 2i -2i 0 0 2 1 1 -2 1 0 -i i i -2i i -i -i 2i -1 -1 -1 2 complex lifted from C4×S3 ρ21 2 -2 0 0 -1 2 -1 -1 -2i 2i 0 0 2 1 1 -2 1 0 i -i -i 2i -i i i -2i -1 -1 -1 2 complex lifted from C4×S3 ρ22 2 -2 0 0 2 -1 -1 -1 -2i 2i 0 0 2 1 -2 1 1 0 i -i 2i -i -i i -2i i 2 -1 -1 -1 complex lifted from C4×S3 ρ23 2 -2 0 0 2 -1 -1 -1 2i -2i 0 0 2 1 -2 1 1 0 -i i -2i i i -i 2i -i 2 -1 -1 -1 complex lifted from C4×S3 ρ24 2 -2 0 0 -1 -1 2 -1 -2i 2i 0 0 2 1 1 1 -2 0 -2i -i -i -i 2i i i i -1 2 -1 -1 complex lifted from C4×S3 ρ25 4 0 -4 0 4 4 4 4 0 0 0 0 -1 0 0 0 0 1 0 0 0 0 0 0 0 0 -1 -1 -1 -1 orthogonal lifted from C2×F5 ρ26 4 0 4 0 4 4 4 4 0 0 0 0 -1 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 -1 -1 -1 orthogonal lifted from F5 ρ27 8 0 0 0 -4 -4 8 -4 0 0 0 0 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 1 -2 1 1 orthogonal lifted from S3×F5 ρ28 8 0 0 0 8 -4 -4 -4 0 0 0 0 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 -2 1 1 1 orthogonal lifted from S3×F5 ρ29 8 0 0 0 -4 8 -4 -4 0 0 0 0 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 -2 orthogonal lifted from S3×F5 ρ30 8 0 0 0 -4 -4 -4 8 0 0 0 0 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 -2 1 orthogonal lifted from S3×F5

Smallest permutation representation of C3⋊S3×F5
On 45 points
Generators in S45
(1 11 6)(2 12 7)(3 13 8)(4 14 9)(5 15 10)(16 26 21)(17 27 22)(18 28 23)(19 29 24)(20 30 25)(31 41 36)(32 42 37)(33 43 38)(34 44 39)(35 45 40)
(1 16 31)(2 17 32)(3 18 33)(4 19 34)(5 20 35)(6 21 36)(7 22 37)(8 23 38)(9 24 39)(10 25 40)(11 26 41)(12 27 42)(13 28 43)(14 29 44)(15 30 45)
(6 11)(7 12)(8 13)(9 14)(10 15)(16 31)(17 32)(18 33)(19 34)(20 35)(21 41)(22 42)(23 43)(24 44)(25 45)(26 36)(27 37)(28 38)(29 39)(30 40)
(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)
(2 3 5 4)(7 8 10 9)(12 13 15 14)(17 18 20 19)(22 23 25 24)(27 28 30 29)(32 33 35 34)(37 38 40 39)(42 43 45 44)

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

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

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

Matrix representation of C3⋊S3×F5 in GL8(𝔽61)

 0 1 0 0 0 0 0 0 60 60 0 0 0 0 0 0 0 0 60 1 0 0 0 0 0 0 60 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 0 0 60 60 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 60 60 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 60 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 60 60 60 60 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0
,
 11 0 0 0 0 0 0 0 0 11 0 0 0 0 0 0 0 0 11 0 0 0 0 0 0 0 0 11 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 60 60 60 60

G:=sub<GL(8,GF(61))| [0,60,0,0,0,0,0,0,1,60,0,0,0,0,0,0,0,0,60,60,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,60,0,0,0,0,0,0,1,60,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,60,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,60,1,0,0,0,0,0,0,60,0,1,0,0,0,0,0,60,0,0,1,0,0,0,0,60,0,0,0],[11,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,1,0,0,60,0,0,0,0,0,0,1,60,0,0,0,0,0,0,0,60,0,0,0,0,0,1,0,60] >;

C3⋊S3×F5 in GAP, Magma, Sage, TeX

C_3\rtimes S_3\times F_5
% in TeX

G:=Group("C3:S3xF5");
// GroupNames label

G:=SmallGroup(360,127);
// by ID

G=gap.SmallGroup(360,127);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-3,-5,24,201,730,5189,2609]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^2=d^5=e^4=1,a*b=b*a,c*a*c=a^-1,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^3>;
// generators/relations

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