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G = C3×C3⋊F5order 180 = 22·32·5

Direct product of C3 and C3⋊F5

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

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

C1C15 — C3×C3⋊F5
C1C5C15C3×D5C32×D5 — C3×C3⋊F5
C15 — C3×C3⋊F5
C1C3

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 >

5C2
2C3
15C4
5C6
5C6
10C6
2C15
5Dic3
15C12
5C3×C6
3F5
2C3×D5
5C3×Dic3
3C3×F5

Character table of C3×C3⋊F5

 class 123A3B3C3D3E4A4B56A6B6C6D6E12A12B12C12D15A15B15C15D15E15F15G15H
 size 151122215154551010101515151544444444
ρ1111111111111111111111111111    trivial
ρ21111111-1-1111111-1-1-1-111111111    linear of order 2
ρ311ζ32ζ3ζ31ζ32111ζ32ζ3ζ31ζ32ζ32ζ3ζ3ζ32ζ31ζ321ζ3ζ32ζ32ζ3    linear of order 3
ρ411ζ3ζ32ζ321ζ3111ζ3ζ32ζ321ζ3ζ3ζ32ζ32ζ3ζ321ζ31ζ32ζ3ζ3ζ32    linear of order 3
ρ511ζ3ζ32ζ321ζ3-1-11ζ3ζ32ζ321ζ3ζ65ζ6ζ6ζ65ζ321ζ31ζ32ζ3ζ3ζ32    linear of order 6
ρ611ζ32ζ3ζ31ζ32-1-11ζ32ζ3ζ31ζ32ζ6ζ65ζ65ζ6ζ31ζ321ζ3ζ32ζ32ζ3    linear of order 6
ρ71-111111i-i1-1-1-1-1-1i-ii-i11111111    linear of order 4
ρ81-111111-ii1-1-1-1-1-1-ii-ii11111111    linear of order 4
ρ91-1ζ3ζ32ζ321ζ3-ii1ζ65ζ6ζ6-1ζ65ζ43ζ3ζ4ζ32ζ43ζ32ζ4ζ3ζ321ζ31ζ32ζ3ζ3ζ32    linear of order 12
ρ101-1ζ32ζ3ζ31ζ32i-i1ζ6ζ65ζ65-1ζ6ζ4ζ32ζ43ζ3ζ4ζ3ζ43ζ32ζ31ζ321ζ3ζ32ζ32ζ3    linear of order 12
ρ111-1ζ3ζ32ζ321ζ3i-i1ζ65ζ6ζ6-1ζ65ζ4ζ3ζ43ζ32ζ4ζ32ζ43ζ3ζ321ζ31ζ32ζ3ζ3ζ32    linear of order 12
ρ121-1ζ32ζ3ζ31ζ32-ii1ζ6ζ65ζ65-1ζ6ζ43ζ32ζ4ζ3ζ43ζ3ζ4ζ32ζ31ζ321ζ3ζ32ζ32ζ3    linear of order 12
ρ132222-1-1-100222-1-1-100002-12-1-1-1-1-1    orthogonal lifted from S3
ρ142-222-1-1-1002-2-211100002-12-1-1-1-1-1    symplectic lifted from Dic3, Schur index 2
ρ152-2-1+-3-1--3ζ6-1ζ650021--31+-3ζ321ζ30000-1--3-1-1+-3-1ζ6ζ65ζ65ζ6    complex lifted from C3×Dic3
ρ162-2-1--3-1+-3ζ65-1ζ60021+-31--3ζ31ζ320000-1+-3-1-1--3-1ζ65ζ6ζ6ζ65    complex lifted from C3×Dic3
ρ1722-1--3-1+-3ζ65-1ζ6002-1--3-1+-3ζ65-1ζ60000-1+-3-1-1--3-1ζ65ζ6ζ6ζ65    complex lifted from C3×S3
ρ1822-1+-3-1--3ζ6-1ζ65002-1+-3-1--3ζ6-1ζ650000-1--3-1-1+-3-1ζ6ζ65ζ65ζ6    complex lifted from C3×S3
ρ19404444400-1000000000-1-1-1-1-1-1-1-1    orthogonal lifted from F5
ρ2040-2-2-3-2+2-3-2+2-34-2-2-300-1000000000ζ65-1ζ6-1ζ65ζ6ζ6ζ65    complex lifted from C3×F5
ρ2140-2+2-3-2-2-3-2-2-34-2+2-300-1000000000ζ6-1ζ65-1ζ6ζ65ζ65ζ6    complex lifted from C3×F5
ρ2240-2+2-3-2-2-31+-3-21--300-1000000000ζ61--15/2ζ651+-15/23ζ533ζ523535232ζ5432ζ53254532ζ5332ζ523253523ζ543ζ53545    complex faithful
ρ2340-2-2-3-2+2-31--3-21+-300-1000000000ζ651--15/2ζ61+-15/232ζ5432ζ5325453ζ533ζ52353523ζ543ζ5354532ζ5332ζ52325352    complex faithful
ρ2440-2+2-3-2-2-31+-3-21--300-1000000000ζ61+-15/2ζ651--15/23ζ543ζ5354532ζ5332ζ5232535232ζ5432ζ5325453ζ533ζ5235352    complex faithful
ρ2540-2-2-3-2+2-31--3-21+-300-1000000000ζ651+-15/2ζ61--15/232ζ5332ζ523253523ζ543ζ535453ζ533ζ523535232ζ5432ζ532545    complex faithful
ρ264044-2-2-200-1000000000-11--15/2-11+-15/21--15/21--15/21+-15/21+-15/2    complex lifted from C3⋊F5
ρ274044-2-2-200-1000000000-11+-15/2-11--15/21+-15/21+-15/21--15/21--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

13000
01300
00130
00013
,
13000
01300
5520470
5520047
,
60100
164400
2171818
44604360
,
43604260
1171718
0010
1010
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

Export

Subgroup lattice of C3×C3⋊F5 in TeX
Character table of C3×C3⋊F5 in TeX

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