non-abelian, supersoluble, monomial
Aliases: He3.2C18, (C3×C27)⋊3S3, C9○He3.2C6, C32.3(S3×C9), C9.6He3⋊3C2, C9.8(C32⋊C6), He3⋊C2.2C9, C3.8(C32⋊C18), He3.4C6.2C3, (C3×C9).24(C3×S3), SmallGroup(486,28)
Series: Derived ►Chief ►Lower central ►Upper central
| He3 — He3.2C18 |
Generators and relations for He3.2C18
G = < a,b,c,d | a3=b3=c3=1, d18=b-1, ab=ba, cac-1=ab-1, dad-1=a-1b, bc=cb, bd=db, dcd-1=a-1c-1 >
Smallest permutation representation of He3.2C18
►On 81 points
Generators in S81
(1 19 10)(2 20 11)(3 21 12)(4 22 13)(5 23 14)(6 24 15)(7 25 16)(8 26 17)(9 27 18)(29 65 47)(31 67 49)(33 69 51)(35 71 53)(37 73 55)(39 75 57)(41 77 59)(43 79 61)(45 81 63)
(1 10 19)(2 11 20)(3 12 21)(4 13 22)(5 14 23)(6 15 24)(7 16 25)(8 17 26)(9 18 27)(28 64 46)(29 65 47)(30 66 48)(31 67 49)(32 68 50)(33 69 51)(34 70 52)(35 71 53)(36 72 54)(37 73 55)(38 74 56)(39 75 57)(40 76 58)(41 77 59)(42 78 60)(43 79 61)(44 80 62)(45 81 63)
(1 45 72)(2 55 28)(3 65 38)(4 75 48)(5 31 58)(6 41 68)(7 51 78)(8 61 34)(9 71 44)(10 81 54)(11 37 64)(12 47 74)(13 57 30)(14 67 40)(15 77 50)(16 33 60)(17 43 70)(18 53 80)(19 63 36)(20 73 46)(21 29 56)(22 39 66)(23 49 76)(24 59 32)(25 69 42)(26 79 52)(27 35 62)
(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 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81)
G:=sub<Sym(81)| (1,19,10)(2,20,11)(3,21,12)(4,22,13)(5,23,14)(6,24,15)(7,25,16)(8,26,17)(9,27,18)(29,65,47)(31,67,49)(33,69,51)(35,71,53)(37,73,55)(39,75,57)(41,77,59)(43,79,61)(45,81,63), (1,10,19)(2,11,20)(3,12,21)(4,13,22)(5,14,23)(6,15,24)(7,16,25)(8,17,26)(9,18,27)(28,64,46)(29,65,47)(30,66,48)(31,67,49)(32,68,50)(33,69,51)(34,70,52)(35,71,53)(36,72,54)(37,73,55)(38,74,56)(39,75,57)(40,76,58)(41,77,59)(42,78,60)(43,79,61)(44,80,62)(45,81,63), (1,45,72)(2,55,28)(3,65,38)(4,75,48)(5,31,58)(6,41,68)(7,51,78)(8,61,34)(9,71,44)(10,81,54)(11,37,64)(12,47,74)(13,57,30)(14,67,40)(15,77,50)(16,33,60)(17,43,70)(18,53,80)(19,63,36)(20,73,46)(21,29,56)(22,39,66)(23,49,76)(24,59,32)(25,69,42)(26,79,52)(27,35,62), (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)(28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81)>;
G:=Group( (1,19,10)(2,20,11)(3,21,12)(4,22,13)(5,23,14)(6,24,15)(7,25,16)(8,26,17)(9,27,18)(29,65,47)(31,67,49)(33,69,51)(35,71,53)(37,73,55)(39,75,57)(41,77,59)(43,79,61)(45,81,63), (1,10,19)(2,11,20)(3,12,21)(4,13,22)(5,14,23)(6,15,24)(7,16,25)(8,17,26)(9,18,27)(28,64,46)(29,65,47)(30,66,48)(31,67,49)(32,68,50)(33,69,51)(34,70,52)(35,71,53)(36,72,54)(37,73,55)(38,74,56)(39,75,57)(40,76,58)(41,77,59)(42,78,60)(43,79,61)(44,80,62)(45,81,63), (1,45,72)(2,55,28)(3,65,38)(4,75,48)(5,31,58)(6,41,68)(7,51,78)(8,61,34)(9,71,44)(10,81,54)(11,37,64)(12,47,74)(13,57,30)(14,67,40)(15,77,50)(16,33,60)(17,43,70)(18,53,80)(19,63,36)(20,73,46)(21,29,56)(22,39,66)(23,49,76)(24,59,32)(25,69,42)(26,79,52)(27,35,62), (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)(28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81) );
G=PermutationGroup([[(1,19,10),(2,20,11),(3,21,12),(4,22,13),(5,23,14),(6,24,15),(7,25,16),(8,26,17),(9,27,18),(29,65,47),(31,67,49),(33,69,51),(35,71,53),(37,73,55),(39,75,57),(41,77,59),(43,79,61),(45,81,63)], [(1,10,19),(2,11,20),(3,12,21),(4,13,22),(5,14,23),(6,15,24),(7,16,25),(8,17,26),(9,18,27),(28,64,46),(29,65,47),(30,66,48),(31,67,49),(32,68,50),(33,69,51),(34,70,52),(35,71,53),(36,72,54),(37,73,55),(38,74,56),(39,75,57),(40,76,58),(41,77,59),(42,78,60),(43,79,61),(44,80,62),(45,81,63)], [(1,45,72),(2,55,28),(3,65,38),(4,75,48),(5,31,58),(6,41,68),(7,51,78),(8,61,34),(9,71,44),(10,81,54),(11,37,64),(12,47,74),(13,57,30),(14,67,40),(15,77,50),(16,33,60),(17,43,70),(18,53,80),(19,63,36),(20,73,46),(21,29,56),(22,39,66),(23,49,76),(24,59,32),(25,69,42),(26,79,52),(27,35,62)], [(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),(28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81)]])
66 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 6A | 6B | 9A | ··· | 9F | 9G | 9H | 9I | 9J | 18A | ··· | 18F | 27A | ··· | 27R | 27S | ··· | 27X | 54A | ··· | 54R |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 6 | 6 | 9 | ··· | 9 | 9 | 9 | 9 | 9 | 18 | ··· | 18 | 27 | ··· | 27 | 27 | ··· | 27 | 54 | ··· | 54 |
| size | 1 | 9 | 1 | 1 | 6 | 18 | 9 | 9 | 1 | ··· | 1 | 6 | 6 | 18 | 18 | 9 | ··· | 9 | 3 | ··· | 3 | 18 | ··· | 18 | 9 | ··· | 9 |
66 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 3 | 6 | 6 |
| type | + | + | + | + | ||||||||
| image | C1 | C2 | C3 | C6 | C9 | C18 | S3 | C3×S3 | S3×C9 | He3.2C18 | C32⋊C6 | C32⋊C18 |
| kernel | He3.2C18 | C9.6He3 | He3.4C6 | C9○He3 | He3⋊C2 | He3 | C3×C27 | C3×C9 | C32 | C1 | C9 | C3 |
| # reps | 1 | 1 | 2 | 2 | 6 | 6 | 1 | 2 | 6 | 36 | 1 | 2 |
Matrix representation of He3.2C18 ►in GL3(𝔽109) generated by
| 45 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 63 |
| 63 | 0 | 0 |
| 0 | 63 | 0 |
| 0 | 0 | 63 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 5 | 0 | 0 |
| 0 | 0 | 7 |
| 0 | 7 | 0 |
G:=sub<GL(3,GF(109))| [45,0,0,0,1,0,0,0,63],[63,0,0,0,63,0,0,0,63],[0,0,1,1,0,0,0,1,0],[5,0,0,0,0,7,0,7,0] >;
He3.2C18 in GAP, Magma, Sage, TeX
{\rm He}_3._2C_{18} % in TeX
G:=Group("He3.2C18"); // GroupNames label
G:=SmallGroup(486,28);
// by ID
G=gap.SmallGroup(486,28);
# by ID
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,43,500,867,873,12964,652]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=c^3=1,d^18=b^-1,a*b=b*a,c*a*c^-1=a*b^-1,d*a*d^-1=a^-1*b,b*c=c*b,b*d=d*b,d*c*d^-1=a^-1*c^-1>;
// generators/relations
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