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## G = C3×C3⋊F5order 180 = 22·32·5

### Direct product of C3 and C3⋊F5

Aliases: C3×C3⋊F5, C151C12, C322F5, C152Dic3, C3⋊(C3×F5), C5⋊(C3×Dic3), (C3×C15)⋊3C4, D5.(C3×S3), (C3×D5).4S3, (C3×D5).1C6, (C32×D5).2C2, SmallGroup(180,21)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C3×C3⋊F5
G = < a,b,c,d | a3=b3=c5=d4=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c3 >

Character table of C3×C3⋊F5

 class 1 2 3A 3B 3C 3D 3E 4A 4B 5 6A 6B 6C 6D 6E 12A 12B 12C 12D 15A 15B 15C 15D 15E 15F 15G 15H size 1 5 1 1 2 2 2 15 15 4 5 5 10 10 10 15 15 15 15 4 4 4 4 4 4 4 4 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 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 linear of order 2 ρ3 1 1 ζ32 ζ3 ζ3 1 ζ32 1 1 1 ζ32 ζ3 ζ3 1 ζ32 ζ32 ζ3 ζ3 ζ32 ζ3 1 ζ32 1 ζ3 ζ32 ζ32 ζ3 linear of order 3 ρ4 1 1 ζ3 ζ32 ζ32 1 ζ3 1 1 1 ζ3 ζ32 ζ32 1 ζ3 ζ3 ζ32 ζ32 ζ3 ζ32 1 ζ3 1 ζ32 ζ3 ζ3 ζ32 linear of order 3 ρ5 1 1 ζ3 ζ32 ζ32 1 ζ3 -1 -1 1 ζ3 ζ32 ζ32 1 ζ3 ζ65 ζ6 ζ6 ζ65 ζ32 1 ζ3 1 ζ32 ζ3 ζ3 ζ32 linear of order 6 ρ6 1 1 ζ32 ζ3 ζ3 1 ζ32 -1 -1 1 ζ32 ζ3 ζ3 1 ζ32 ζ6 ζ65 ζ65 ζ6 ζ3 1 ζ32 1 ζ3 ζ32 ζ32 ζ3 linear of order 6 ρ7 1 -1 1 1 1 1 1 i -i 1 -1 -1 -1 -1 -1 i -i i -i 1 1 1 1 1 1 1 1 linear of order 4 ρ8 1 -1 1 1 1 1 1 -i i 1 -1 -1 -1 -1 -1 -i i -i i 1 1 1 1 1 1 1 1 linear of order 4 ρ9 1 -1 ζ3 ζ32 ζ32 1 ζ3 -i i 1 ζ65 ζ6 ζ6 -1 ζ65 ζ43ζ3 ζ4ζ32 ζ43ζ32 ζ4ζ3 ζ32 1 ζ3 1 ζ32 ζ3 ζ3 ζ32 linear of order 12 ρ10 1 -1 ζ32 ζ3 ζ3 1 ζ32 i -i 1 ζ6 ζ65 ζ65 -1 ζ6 ζ4ζ32 ζ43ζ3 ζ4ζ3 ζ43ζ32 ζ3 1 ζ32 1 ζ3 ζ32 ζ32 ζ3 linear of order 12 ρ11 1 -1 ζ3 ζ32 ζ32 1 ζ3 i -i 1 ζ65 ζ6 ζ6 -1 ζ65 ζ4ζ3 ζ43ζ32 ζ4ζ32 ζ43ζ3 ζ32 1 ζ3 1 ζ32 ζ3 ζ3 ζ32 linear of order 12 ρ12 1 -1 ζ32 ζ3 ζ3 1 ζ32 -i i 1 ζ6 ζ65 ζ65 -1 ζ6 ζ43ζ32 ζ4ζ3 ζ43ζ3 ζ4ζ32 ζ3 1 ζ32 1 ζ3 ζ32 ζ32 ζ3 linear of order 12 ρ13 2 2 2 2 -1 -1 -1 0 0 2 2 2 -1 -1 -1 0 0 0 0 2 -1 2 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ14 2 -2 2 2 -1 -1 -1 0 0 2 -2 -2 1 1 1 0 0 0 0 2 -1 2 -1 -1 -1 -1 -1 symplectic lifted from Dic3, Schur index 2 ρ15 2 -2 -1+√-3 -1-√-3 ζ6 -1 ζ65 0 0 2 1-√-3 1+√-3 ζ32 1 ζ3 0 0 0 0 -1-√-3 -1 -1+√-3 -1 ζ6 ζ65 ζ65 ζ6 complex lifted from C3×Dic3 ρ16 2 -2 -1-√-3 -1+√-3 ζ65 -1 ζ6 0 0 2 1+√-3 1-√-3 ζ3 1 ζ32 0 0 0 0 -1+√-3 -1 -1-√-3 -1 ζ65 ζ6 ζ6 ζ65 complex lifted from C3×Dic3 ρ17 2 2 -1-√-3 -1+√-3 ζ65 -1 ζ6 0 0 2 -1-√-3 -1+√-3 ζ65 -1 ζ6 0 0 0 0 -1+√-3 -1 -1-√-3 -1 ζ65 ζ6 ζ6 ζ65 complex lifted from C3×S3 ρ18 2 2 -1+√-3 -1-√-3 ζ6 -1 ζ65 0 0 2 -1+√-3 -1-√-3 ζ6 -1 ζ65 0 0 0 0 -1-√-3 -1 -1+√-3 -1 ζ6 ζ65 ζ65 ζ6 complex lifted from C3×S3 ρ19 4 0 4 4 4 4 4 0 0 -1 0 0 0 0 0 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from F5 ρ20 4 0 -2-2√-3 -2+2√-3 -2+2√-3 4 -2-2√-3 0 0 -1 0 0 0 0 0 0 0 0 0 ζ65 -1 ζ6 -1 ζ65 ζ6 ζ6 ζ65 complex lifted from C3×F5 ρ21 4 0 -2+2√-3 -2-2√-3 -2-2√-3 4 -2+2√-3 0 0 -1 0 0 0 0 0 0 0 0 0 ζ6 -1 ζ65 -1 ζ6 ζ65 ζ65 ζ6 complex lifted from C3×F5 ρ22 4 0 -2+2√-3 -2-2√-3 1+√-3 -2 1-√-3 0 0 -1 0 0 0 0 0 0 0 0 0 ζ6 1-√-15/2 ζ65 1+√-15/2 -ζ3ζ53-ζ3ζ52-ζ3+ζ53+ζ52 -ζ32ζ54-ζ32ζ5-ζ32+ζ54+ζ5 -ζ32ζ53-ζ32ζ52-ζ32+ζ53+ζ52 -ζ3ζ54-ζ3ζ5-ζ3+ζ54+ζ5 complex faithful ρ23 4 0 -2-2√-3 -2+2√-3 1-√-3 -2 1+√-3 0 0 -1 0 0 0 0 0 0 0 0 0 ζ65 1-√-15/2 ζ6 1+√-15/2 -ζ32ζ54-ζ32ζ5-ζ32+ζ54+ζ5 -ζ3ζ53-ζ3ζ52-ζ3+ζ53+ζ52 -ζ3ζ54-ζ3ζ5-ζ3+ζ54+ζ5 -ζ32ζ53-ζ32ζ52-ζ32+ζ53+ζ52 complex faithful ρ24 4 0 -2+2√-3 -2-2√-3 1+√-3 -2 1-√-3 0 0 -1 0 0 0 0 0 0 0 0 0 ζ6 1+√-15/2 ζ65 1-√-15/2 -ζ3ζ54-ζ3ζ5-ζ3+ζ54+ζ5 -ζ32ζ53-ζ32ζ52-ζ32+ζ53+ζ52 -ζ32ζ54-ζ32ζ5-ζ32+ζ54+ζ5 -ζ3ζ53-ζ3ζ52-ζ3+ζ53+ζ52 complex faithful ρ25 4 0 -2-2√-3 -2+2√-3 1-√-3 -2 1+√-3 0 0 -1 0 0 0 0 0 0 0 0 0 ζ65 1+√-15/2 ζ6 1-√-15/2 -ζ32ζ53-ζ32ζ52-ζ32+ζ53+ζ52 -ζ3ζ54-ζ3ζ5-ζ3+ζ54+ζ5 -ζ3ζ53-ζ3ζ52-ζ3+ζ53+ζ52 -ζ32ζ54-ζ32ζ5-ζ32+ζ54+ζ5 complex faithful ρ26 4 0 4 4 -2 -2 -2 0 0 -1 0 0 0 0 0 0 0 0 0 -1 1-√-15/2 -1 1+√-15/2 1-√-15/2 1-√-15/2 1+√-15/2 1+√-15/2 complex lifted from C3⋊F5 ρ27 4 0 4 4 -2 -2 -2 0 0 -1 0 0 0 0 0 0 0 0 0 -1 1+√-15/2 -1 1-√-15/2 1+√-15/2 1+√-15/2 1-√-15/2 1-√-15/2 complex lifted from C3⋊F5

Permutation representations of C3×C3⋊F5
On 30 points - transitive group 30T47
Generators in S30
(1 9 14)(2 10 15)(3 6 11)(4 7 12)(5 8 13)(16 21 26)(17 22 27)(18 23 28)(19 24 29)(20 25 30)
(1 9 14)(2 10 15)(3 6 11)(4 7 12)(5 8 13)(16 26 21)(17 27 22)(18 28 23)(19 29 24)(20 30 25)
(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 18)(2 20 5 16)(3 17 4 19)(6 22 7 24)(8 21 10 25)(9 23)(11 27 12 29)(13 26 15 30)(14 28)

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

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

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

G:=TransitiveGroup(30,47);

C3×C3⋊F5 is a maximal subgroup of   C3×S3×F5  C3⋊F5⋊S3

Matrix representation of C3×C3⋊F5 in GL4(𝔽61) generated by

 13 0 0 0 0 13 0 0 0 0 13 0 0 0 0 13
,
 13 0 0 0 0 13 0 0 55 20 47 0 55 20 0 47
,
 60 1 0 0 16 44 0 0 2 17 18 18 44 60 43 60
,
 43 60 42 60 1 17 17 18 0 0 1 0 1 0 1 0
G:=sub<GL(4,GF(61))| [13,0,0,0,0,13,0,0,0,0,13,0,0,0,0,13],[13,0,55,55,0,13,20,20,0,0,47,0,0,0,0,47],[60,16,2,44,1,44,17,60,0,0,18,43,0,0,18,60],[43,1,0,1,60,17,0,0,42,17,1,1,60,18,0,0] >;

C3×C3⋊F5 in GAP, Magma, Sage, TeX

C_3\times C_3\rtimes F_5
% in TeX

G:=Group("C3xC3:F5");
// GroupNames label

G:=SmallGroup(180,21);
// by ID

G=gap.SmallGroup(180,21);
# by ID

G:=PCGroup([5,-2,-3,-2,-3,-5,30,483,2704,614]);
// Polycyclic

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

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