non-abelian, supersoluble, monomial
Aliases: He3⋊4D9, (C9×He3)⋊5C2, C9⋊(He3⋊C2), C32⋊C9⋊16S3, C32⋊2(C9⋊S3), (C32×C9)⋊18S3, (C3×He3).23S3, C33.37(C3⋊S3), C3.2(He3⋊5S3), C3.4(C32⋊4D9), C32.11(C33⋊C2), (C3×C9).21(C3⋊S3), SmallGroup(486,182)
Series: Derived ►Chief ►Lower central ►Upper central
| C9×He3 — He3⋊4D9 |
Generators and relations for He3⋊4D9
G = < a,b,c,d,e | a3=b3=c3=d9=e2=1, ab=ba, cac-1=ab-1, ad=da, eae=a-1, bc=cb, bd=db, be=eb, cd=dc, ece=c-1, ede=d-1 >
Subgroups: 1420 in 172 conjugacy classes, 54 normal (9 characteristic)
C1, C2, C3, C3, S3, C6, C9, C9, C32, C32, C32, D9, C3×S3, C3⋊S3, C3×C9, C3×C9, C3×C9, He3, C33, C3×D9, C9⋊S3, He3⋊C2, C3×C3⋊S3, C32⋊C9, C32×C9, C3×He3, C32⋊2D9, C3×C9⋊S3, He3⋊5S3, C9×He3, He3⋊4D9
Quotients: C1, C2, S3, D9, C3⋊S3, C9⋊S3, He3⋊C2, C33⋊C2, C32⋊4D9, He3⋊5S3, He3⋊4D9
►On 54 points
Generators in S54
(1 33 42)(2 34 43)(3 35 44)(4 36 45)(5 28 37)(6 29 38)(7 30 39)(8 31 40)(9 32 41)(10 47 27)(11 48 19)(12 49 20)(13 50 21)(14 51 22)(15 52 23)(16 53 24)(17 54 25)(18 46 26)
(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 45 36)(2 37 28)(3 38 29)(4 39 30)(5 40 31)(6 41 32)(7 42 33)(8 43 34)(9 44 35)(10 24 53)(11 25 54)(12 26 46)(13 27 47)(14 19 48)(15 20 49)(16 21 50)(17 22 51)(18 23 52)
(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 13)(2 12)(3 11)(4 10)(5 18)(6 17)(7 16)(8 15)(9 14)(19 35)(20 34)(21 33)(22 32)(23 31)(24 30)(25 29)(26 28)(27 36)(37 46)(38 54)(39 53)(40 52)(41 51)(42 50)(43 49)(44 48)(45 47)
G:=sub<Sym(54)| (1,33,42)(2,34,43)(3,35,44)(4,36,45)(5,28,37)(6,29,38)(7,30,39)(8,31,40)(9,32,41)(10,47,27)(11,48,19)(12,49,20)(13,50,21)(14,51,22)(15,52,23)(16,53,24)(17,54,25)(18,46,26), (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), (1,45,36)(2,37,28)(3,38,29)(4,39,30)(5,40,31)(6,41,32)(7,42,33)(8,43,34)(9,44,35)(10,24,53)(11,25,54)(12,26,46)(13,27,47)(14,19,48)(15,20,49)(16,21,50)(17,22,51)(18,23,52), (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), (1,13)(2,12)(3,11)(4,10)(5,18)(6,17)(7,16)(8,15)(9,14)(19,35)(20,34)(21,33)(22,32)(23,31)(24,30)(25,29)(26,28)(27,36)(37,46)(38,54)(39,53)(40,52)(41,51)(42,50)(43,49)(44,48)(45,47)>;
G:=Group( (1,33,42)(2,34,43)(3,35,44)(4,36,45)(5,28,37)(6,29,38)(7,30,39)(8,31,40)(9,32,41)(10,47,27)(11,48,19)(12,49,20)(13,50,21)(14,51,22)(15,52,23)(16,53,24)(17,54,25)(18,46,26), (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), (1,45,36)(2,37,28)(3,38,29)(4,39,30)(5,40,31)(6,41,32)(7,42,33)(8,43,34)(9,44,35)(10,24,53)(11,25,54)(12,26,46)(13,27,47)(14,19,48)(15,20,49)(16,21,50)(17,22,51)(18,23,52), (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), (1,13)(2,12)(3,11)(4,10)(5,18)(6,17)(7,16)(8,15)(9,14)(19,35)(20,34)(21,33)(22,32)(23,31)(24,30)(25,29)(26,28)(27,36)(37,46)(38,54)(39,53)(40,52)(41,51)(42,50)(43,49)(44,48)(45,47) );
G=PermutationGroup([[(1,33,42),(2,34,43),(3,35,44),(4,36,45),(5,28,37),(6,29,38),(7,30,39),(8,31,40),(9,32,41),(10,47,27),(11,48,19),(12,49,20),(13,50,21),(14,51,22),(15,52,23),(16,53,24),(17,54,25),(18,46,26)], [(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)], [(1,45,36),(2,37,28),(3,38,29),(4,39,30),(5,40,31),(6,41,32),(7,42,33),(8,43,34),(9,44,35),(10,24,53),(11,25,54),(12,26,46),(13,27,47),(14,19,48),(15,20,49),(16,21,50),(17,22,51),(18,23,52)], [(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)], [(1,13),(2,12),(3,11),(4,10),(5,18),(6,17),(7,16),(8,15),(9,14),(19,35),(20,34),(21,33),(22,32),(23,31),(24,30),(25,29),(26,28),(27,36),(37,46),(38,54),(39,53),(40,52),(41,51),(42,50),(43,49),(44,48),(45,47)]])
54 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 3F | ··· | 3Q | 6A | 6B | 9A | ··· | 9I | 9J | ··· | 9AG |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | ··· | 3 | 6 | 6 | 9 | ··· | 9 | 9 | ··· | 9 |
| size | 1 | 81 | 1 | 1 | 2 | 2 | 2 | 6 | ··· | 6 | 81 | 81 | 2 | ··· | 2 | 6 | ··· | 6 |
54 irreducible representations
| dim | 1 | 1 | 2 | 2 | 2 | 2 | 3 | 6 | 6 |
| type | + | + | + | + | + | + | |||
| image | C1 | C2 | S3 | S3 | S3 | D9 | He3⋊C2 | He3⋊5S3 | He3⋊4D9 |
| kernel | He3⋊4D9 | C9×He3 | C32⋊C9 | C32×C9 | C3×He3 | He3 | C9 | C3 | C1 |
| # reps | 1 | 1 | 8 | 4 | 1 | 27 | 4 | 2 | 6 |
Matrix representation of He3⋊4D9 ►in GL5(𝔽19)
| 17 | 16 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 7 | 0 |
| 0 | 0 | 0 | 0 | 11 |
| 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 | 1 |
| 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 16 | 4 | 0 | 0 | 0 |
| 5 | 12 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 18 | 18 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 |
| 0 | 0 | 0 | 12 | 0 |
G:=sub<GL(5,GF(19))| [17,1,0,0,0,16,1,0,0,0,0,0,1,0,0,0,0,0,7,0,0,0,0,0,11],[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],[1,0,0,0,0,0,1,0,0,0,0,0,0,8,0,0,0,0,0,12,0,0,1,0,0],[16,5,0,0,0,4,12,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,18,0,0,0,0,18,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,8,0] >;
He3⋊4D9 in GAP, Magma, Sage, TeX
{\rm He}_3\rtimes_4D_9 % in TeX
G:=Group("He3:4D9"); // GroupNames label
G:=SmallGroup(486,182);
// by ID
G=gap.SmallGroup(486,182);
# by ID
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,697,655,218,867,735,3244]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^9=e^2=1,a*b=b*a,c*a*c^-1=a*b^-1,a*d=d*a,e*a*e=a^-1,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e=c^-1,e*d*e=d^-1>;
// generators/relations