non-abelian, supersoluble, monomial
Aliases: (C32×C9)⋊9S3, He3.C3⋊5S3, He3.9(C3×S3), C3⋊(He3.C6), (C3×He3).11C6, C33.44(C3×S3), He3⋊5S3.4C3, C3.14(He3⋊4S3), C32.12(C32⋊C6), (C3×C9)⋊2(C3⋊S3), (C3×He3.C3)⋊7C2, C32.4(C3×C3⋊S3), SmallGroup(486,169)
Series: Derived ►Chief ►Lower central ►Upper central
| C3×He3 — He3.C3⋊S3 |
Generators and relations for He3.C3⋊S3
G = < a,b,c,d,e,f | a3=b3=c3=e3=f2=1, d3=b, ab=ba, cac-1=ab-1, ad=da, ae=ea, faf=a-1b, bc=cb, bd=db, be=eb, bf=fb, dcd-1=ab-1c, ce=ec, fcf=bc-1, de=ed, df=fd, fef=e-1 >
Subgroups: 596 in 90 conjugacy classes, 20 normal (12 characteristic)
C1, C2, C3, C3, S3, C6, C9, C32, C32, C32, C18, C3×S3, C3⋊S3, C3×C9, C3×C9, He3, He3, 3- 1+2, C33, C33, S3×C9, He3⋊C2, C3×C3⋊S3, He3.C3, He3.C3, C32×C9, C3×He3, C3×3- 1+2, He3.C6, C9×C3⋊S3, He3⋊5S3, C3×He3.C3, He3.C3⋊S3
Quotients: C1, C2, C3, S3, C6, C3×S3, C3⋊S3, C32⋊C6, C3×C3⋊S3, He3.C6, He3⋊4S3, He3.C3⋊S3
►On 54 points
Generators in S54
(1 4 7)(2 5 8)(3 6 9)(10 13 16)(11 14 17)(12 15 18)(19 25 22)(20 26 23)(21 27 24)(37 43 40)(38 44 41)(39 45 42)
(1 4 7)(2 5 8)(3 6 9)(10 13 16)(11 14 17)(12 15 18)(19 22 25)(20 23 26)(21 24 27)(28 31 34)(29 32 35)(30 33 36)(37 40 43)(38 41 44)(39 42 45)(46 49 52)(47 50 53)(48 51 54)
(1 33 42)(2 34 40)(3 35 38)(4 36 45)(5 28 43)(6 29 41)(7 30 39)(8 31 37)(9 32 44)(10 53 22)(11 54 20)(12 46 27)(13 47 25)(14 48 23)(15 49 21)(16 50 19)(17 51 26)(18 52 24)
(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)(46 47 48 49 50 51 52 53 54)
(1 4 7)(2 5 8)(3 6 9)(10 16 13)(11 17 14)(12 18 15)(19 25 22)(20 26 23)(21 27 24)(28 31 34)(29 32 35)(30 33 36)(37 40 43)(38 41 44)(39 42 45)(46 52 49)(47 53 50)(48 54 51)
(1 49)(2 50)(3 51)(4 52)(5 53)(6 54)(7 46)(8 47)(9 48)(10 34)(11 35)(12 36)(13 28)(14 29)(15 30)(16 31)(17 32)(18 33)(19 43)(20 44)(21 45)(22 37)(23 38)(24 39)(25 40)(26 41)(27 42)
G:=sub<Sym(54)| (1,4,7)(2,5,8)(3,6,9)(10,13,16)(11,14,17)(12,15,18)(19,25,22)(20,26,23)(21,27,24)(37,43,40)(38,44,41)(39,45,42), (1,4,7)(2,5,8)(3,6,9)(10,13,16)(11,14,17)(12,15,18)(19,22,25)(20,23,26)(21,24,27)(28,31,34)(29,32,35)(30,33,36)(37,40,43)(38,41,44)(39,42,45)(46,49,52)(47,50,53)(48,51,54), (1,33,42)(2,34,40)(3,35,38)(4,36,45)(5,28,43)(6,29,41)(7,30,39)(8,31,37)(9,32,44)(10,53,22)(11,54,20)(12,46,27)(13,47,25)(14,48,23)(15,49,21)(16,50,19)(17,51,26)(18,52,24), (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)(46,47,48,49,50,51,52,53,54), (1,4,7)(2,5,8)(3,6,9)(10,16,13)(11,17,14)(12,18,15)(19,25,22)(20,26,23)(21,27,24)(28,31,34)(29,32,35)(30,33,36)(37,40,43)(38,41,44)(39,42,45)(46,52,49)(47,53,50)(48,54,51), (1,49)(2,50)(3,51)(4,52)(5,53)(6,54)(7,46)(8,47)(9,48)(10,34)(11,35)(12,36)(13,28)(14,29)(15,30)(16,31)(17,32)(18,33)(19,43)(20,44)(21,45)(22,37)(23,38)(24,39)(25,40)(26,41)(27,42)>;
G:=Group( (1,4,7)(2,5,8)(3,6,9)(10,13,16)(11,14,17)(12,15,18)(19,25,22)(20,26,23)(21,27,24)(37,43,40)(38,44,41)(39,45,42), (1,4,7)(2,5,8)(3,6,9)(10,13,16)(11,14,17)(12,15,18)(19,22,25)(20,23,26)(21,24,27)(28,31,34)(29,32,35)(30,33,36)(37,40,43)(38,41,44)(39,42,45)(46,49,52)(47,50,53)(48,51,54), (1,33,42)(2,34,40)(3,35,38)(4,36,45)(5,28,43)(6,29,41)(7,30,39)(8,31,37)(9,32,44)(10,53,22)(11,54,20)(12,46,27)(13,47,25)(14,48,23)(15,49,21)(16,50,19)(17,51,26)(18,52,24), (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)(46,47,48,49,50,51,52,53,54), (1,4,7)(2,5,8)(3,6,9)(10,16,13)(11,17,14)(12,18,15)(19,25,22)(20,26,23)(21,27,24)(28,31,34)(29,32,35)(30,33,36)(37,40,43)(38,41,44)(39,42,45)(46,52,49)(47,53,50)(48,54,51), (1,49)(2,50)(3,51)(4,52)(5,53)(6,54)(7,46)(8,47)(9,48)(10,34)(11,35)(12,36)(13,28)(14,29)(15,30)(16,31)(17,32)(18,33)(19,43)(20,44)(21,45)(22,37)(23,38)(24,39)(25,40)(26,41)(27,42) );
G=PermutationGroup([[(1,4,7),(2,5,8),(3,6,9),(10,13,16),(11,14,17),(12,15,18),(19,25,22),(20,26,23),(21,27,24),(37,43,40),(38,44,41),(39,45,42)], [(1,4,7),(2,5,8),(3,6,9),(10,13,16),(11,14,17),(12,15,18),(19,22,25),(20,23,26),(21,24,27),(28,31,34),(29,32,35),(30,33,36),(37,40,43),(38,41,44),(39,42,45),(46,49,52),(47,50,53),(48,51,54)], [(1,33,42),(2,34,40),(3,35,38),(4,36,45),(5,28,43),(6,29,41),(7,30,39),(8,31,37),(9,32,44),(10,53,22),(11,54,20),(12,46,27),(13,47,25),(14,48,23),(15,49,21),(16,50,19),(17,51,26),(18,52,24)], [(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),(46,47,48,49,50,51,52,53,54)], [(1,4,7),(2,5,8),(3,6,9),(10,16,13),(11,17,14),(12,18,15),(19,25,22),(20,26,23),(21,27,24),(28,31,34),(29,32,35),(30,33,36),(37,40,43),(38,41,44),(39,42,45),(46,52,49),(47,53,50),(48,54,51)], [(1,49),(2,50),(3,51),(4,52),(5,53),(6,54),(7,46),(8,47),(9,48),(10,34),(11,35),(12,36),(13,28),(14,29),(15,30),(16,31),(17,32),(18,33),(19,43),(20,44),(21,45),(22,37),(23,38),(24,39),(25,40),(26,41),(27,42)]])
39 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 3F | 3G | 3H | 3I | 3J | 3K | 6A | 6B | 9A | ··· | 9F | 9G | ··· | 9L | 9M | ··· | 9R | 18A | ··· | 18F |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 6 | 6 | 9 | ··· | 9 | 9 | ··· | 9 | 9 | ··· | 9 | 18 | ··· | 18 |
| size | 1 | 27 | 1 | 1 | 2 | 2 | 2 | 6 | 6 | 6 | 18 | 18 | 18 | 27 | 27 | 3 | ··· | 3 | 6 | ··· | 6 | 18 | ··· | 18 | 27 | ··· | 27 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 3 | 6 | 6 |
| type | + | + | + | + | + | ||||||
| image | C1 | C2 | C3 | C6 | S3 | S3 | C3×S3 | C3×S3 | He3.C6 | C32⋊C6 | He3.C3⋊S3 |
| kernel | He3.C3⋊S3 | C3×He3.C3 | He3⋊5S3 | C3×He3 | He3.C3 | C32×C9 | He3 | C33 | C3 | C32 | C1 |
| # reps | 1 | 1 | 2 | 2 | 3 | 1 | 6 | 2 | 12 | 3 | 6 |
Matrix representation of He3.C3⋊S3 ►in GL5(𝔽19)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 11 | 0 |
| 0 | 0 | 0 | 0 | 7 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 11 | 0 | 0 |
| 0 | 0 | 0 | 11 | 0 |
| 0 | 0 | 0 | 0 | 11 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 7 |
| 0 | 0 | 11 | 0 | 0 |
| 7 | 0 | 0 | 0 | 0 |
| 0 | 7 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 5 |
| 0 | 18 | 0 | 0 | 0 |
| 1 | 18 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 18 | 0 | 0 | 0 |
| 18 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 18 |
G:=sub<GL(5,GF(19))| [1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,11,0,0,0,0,0,7],[1,0,0,0,0,0,1,0,0,0,0,0,11,0,0,0,0,0,11,0,0,0,0,0,11],[1,0,0,0,0,0,1,0,0,0,0,0,0,0,11,0,0,1,0,0,0,0,0,7,0],[7,0,0,0,0,0,7,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,5],[0,1,0,0,0,18,18,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[0,18,0,0,0,18,0,0,0,0,0,0,0,12,0,0,0,8,0,0,0,0,0,0,18] >;
He3.C3⋊S3 in GAP, Magma, Sage, TeX
{\rm He}_3.C_3\rtimes S_3 % in TeX
G:=Group("He3.C3:S3"); // GroupNames label
G:=SmallGroup(486,169);
// by ID
G=gap.SmallGroup(486,169);
# by ID
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,979,218,548,867,12964,1906]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^3=b^3=c^3=e^3=f^2=1,d^3=b,a*b=b*a,c*a*c^-1=a*b^-1,a*d=d*a,a*e=e*a,f*a*f=a^-1*b,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,d*c*d^-1=a*b^-1*c,c*e=e*c,f*c*f=b*c^-1,d*e=e*d,d*f=f*d,f*e*f=e^-1>;
// generators/relations