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G = C3:S3xF5order 360 = 23·32·5

Direct product of C3:S3 and F5

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

Aliases: C3:S3xF5, (C3xF5):S3, C3:1(S3xF5), C15:1(C4xS3), C3:D15:2C4, C32:6(C2xF5), (C3xD5).2D6, (C32xF5):2C2, C32:3F5:1C2, (C32xD5).2C22, C5:(C4xC3:S3), (C5xC3:S3):2C4, (C3xC15):4(C2xC4), D5.1(C2xC3:S3), (D5xC3:S3).3C2, SmallGroup(360,127)

Series: Derived Chief Lower central Upper central

C1C3xC15 — C3:S3xF5
C1C5C15C3xC15C32xD5C32xF5 — C3:S3xF5
C3xC15 — C3:S3xF5
C1

Generators and relations for C3:S3xF5
 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, C2xC4, C32, D5, D5, C10, Dic3, C12, D6, C15, C3:S3, C3:S3, C3xC6, F5, F5, D10, C4xS3, C5xS3, C3xD5, D15, C3:Dic3, C3xC12, C2xC3:S3, C2xF5, C3xC15, C3xF5, C3:F5, S3xD5, C4xC3:S3, C32xD5, C5xC3:S3, C3:D15, S3xF5, C32xF5, C32:3F5, D5xC3:S3, C3:S3xF5
Quotients: C1, C2, C4, C22, S3, C2xC4, D6, C3:S3, F5, C4xS3, C2xC3:S3, C2xF5, C4xC3:S3, S3xF5, C3:S3xF5

Character table of C3:S3xF5

 class 12A2B2C3A3B3C3D4A4B4C4D56A6B6C6D1012A12B12C12D12E12F12G12H15A15B15C15D
 size 1594522225545454101010103610101010101010108888
ρ1111111111111111111111111111111    trivial
ρ211-1-1111111-1-111111-1111111111111    linear of order 2
ρ311111111-1-1-1-1111111-1-1-1-1-1-1-1-11111    linear of order 2
ρ411-1-11111-1-11111111-1-1-1-1-1-1-1-1-11111    linear of order 2
ρ51-1-111111i-i-ii1-1-1-1-1-1i-i-i-i-iiii1111    linear of order 4
ρ61-11-11111i-ii-i1-1-1-1-11i-i-i-i-iiii1111    linear of order 4
ρ71-11-11111-ii-ii1-1-1-1-11-iiiii-i-i-i1111    linear of order 4
ρ81-1-111111-iii-i1-1-1-1-1-1-iiiii-i-i-i1111    linear of order 4
ρ922002-1-1-1-2-2002-12-1-1011-2111-212-1-1-1    orthogonal lifted from D6
ρ102200-1-1-12-2-20022-1-1-101-2111-211-1-12-1    orthogonal lifted from D6
ρ112200-12-1-122002-1-12-10-1-1-12-1-1-12-1-1-12    orthogonal lifted from S3
ρ122200-1-12-1-2-2002-1-1-120-2111-2111-12-1-1    orthogonal lifted from D6
ρ132200-1-12-122002-1-1-1202-1-1-12-1-1-1-12-1-1    orthogonal lifted from S3
ρ1422002-1-1-122002-12-1-10-1-12-1-1-12-12-1-1-1    orthogonal lifted from S3
ρ152200-12-1-1-2-2002-1-12-10111-2111-2-1-1-12    orthogonal lifted from D6
ρ162200-1-1-12220022-1-1-10-12-1-1-12-1-1-1-12-1    orthogonal lifted from S3
ρ172-200-1-12-12i-2i002111-202iiii-2i-i-i-i-12-1-1    complex lifted from C4xS3
ρ182-200-1-1-12-2i2i002-21110i2i-i-i-i-2iii-1-12-1    complex lifted from C4xS3
ρ192-200-1-1-122i-2i002-21110-i-2iiii2i-i-i-1-12-1    complex lifted from C4xS3
ρ202-200-12-1-12i-2i00211-210-iii-2ii-i-i2i-1-1-12    complex lifted from C4xS3
ρ212-200-12-1-1-2i2i00211-210i-i-i2i-iii-2i-1-1-12    complex lifted from C4xS3
ρ222-2002-1-1-1-2i2i0021-2110i-i2i-i-ii-2ii2-1-1-1    complex lifted from C4xS3
ρ232-2002-1-1-12i-2i0021-2110-ii-2iii-i2i-i2-1-1-1    complex lifted from C4xS3
ρ242-200-1-12-1-2i2i002111-20-2i-i-i-i2iiii-12-1-1    complex lifted from C4xS3
ρ2540-4044440000-10000100000000-1-1-1-1    orthogonal lifted from C2xF5
ρ26404044440000-10000-100000000-1-1-1-1    orthogonal lifted from F5
ρ278000-4-48-40000-200000000000001-211    orthogonal lifted from S3xF5
ρ2880008-4-4-40000-20000000000000-2111    orthogonal lifted from S3xF5
ρ298000-48-4-40000-20000000000000111-2    orthogonal lifted from S3xF5
ρ308000-4-4-480000-2000000000000011-21    orthogonal lifted from S3xF5

Smallest permutation representation of C3:S3xF5
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:S3xF5 in GL8(F61)

01000000
6060000000
006010000
006000000
00001000
00000100
00000010
00000001
,
01000000
6060000000
00100000
00010000
00001000
00000100
00000010
00000001
,
10000000
6060000000
00100000
001600000
00001000
00000100
00000010
00000001
,
10000000
01000000
00100000
00010000
000060606060
00001000
00000100
00000010
,
110000000
011000000
001100000
000110000
00001000
00000001
00000100
000060606060

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:S3xF5 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

Export

Character table of C3:S3xF5 in TeX

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