non-abelian, supersoluble, monomial
Aliases: He3.1C18, (C3xC27):2S3, C9oHe3.1C6, C32.2(S3xC9), C9.5He3:3C2, C9.7(C32:C6), He3:C2.1C9, C3.7(C32:C18), He3.4C6.1C3, (C3xC9).23(C3xS3), SmallGroup(486,26)
Series: Derived ►Chief ►Lower central ►Upper central
| He3 — He3.C18 |
Generators and relations for He3.C18
G = < a,b,c,d | a3=b3=c3=1, d18=b, ab=ba, cac-1=ab-1, dad-1=a-1b, bc=cb, bd=db, dcd-1=a-1c-1 >
Quotients: C1, C2, C3, S3, C6, C9, C18, C3xS3, S3xC9, C32:C6, C32:C18, He3.C18
Smallest permutation representation of He3.C18
►On 81 points
Generators in S81
(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 46 64)(30 48 66)(32 50 68)(34 52 70)(36 54 72)(38 56 74)(40 58 76)(42 60 78)(44 62 80)
(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)(28 46 64)(29 47 65)(30 48 66)(31 49 67)(32 50 68)(33 51 69)(34 52 70)(35 53 71)(36 54 72)(37 55 73)(38 56 74)(39 57 75)(40 58 76)(41 59 77)(42 60 78)(43 61 79)(44 62 80)(45 63 81)
(1 58 31)(2 50 77)(3 42 69)(4 34 61)(5 80 53)(6 72 45)(7 64 37)(8 56 29)(9 48 75)(10 40 67)(11 32 59)(12 78 51)(13 70 43)(14 62 35)(15 54 81)(16 46 73)(17 38 65)(18 30 57)(19 76 49)(20 68 41)(21 60 33)(22 52 79)(23 44 71)(24 36 63)(25 28 55)(26 74 47)(27 66 39)
(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)| (1,10,19)(2,11,20)(3,12,21)(4,13,22)(5,14,23)(6,15,24)(7,16,25)(8,17,26)(9,18,27)(28,46,64)(30,48,66)(32,50,68)(34,52,70)(36,54,72)(38,56,74)(40,58,76)(42,60,78)(44,62,80), (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)(28,46,64)(29,47,65)(30,48,66)(31,49,67)(32,50,68)(33,51,69)(34,52,70)(35,53,71)(36,54,72)(37,55,73)(38,56,74)(39,57,75)(40,58,76)(41,59,77)(42,60,78)(43,61,79)(44,62,80)(45,63,81), (1,58,31)(2,50,77)(3,42,69)(4,34,61)(5,80,53)(6,72,45)(7,64,37)(8,56,29)(9,48,75)(10,40,67)(11,32,59)(12,78,51)(13,70,43)(14,62,35)(15,54,81)(16,46,73)(17,38,65)(18,30,57)(19,76,49)(20,68,41)(21,60,33)(22,52,79)(23,44,71)(24,36,63)(25,28,55)(26,74,47)(27,66,39), (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( (1,10,19)(2,11,20)(3,12,21)(4,13,22)(5,14,23)(6,15,24)(7,16,25)(8,17,26)(9,18,27)(28,46,64)(30,48,66)(32,50,68)(34,52,70)(36,54,72)(38,56,74)(40,58,76)(42,60,78)(44,62,80), (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)(28,46,64)(29,47,65)(30,48,66)(31,49,67)(32,50,68)(33,51,69)(34,52,70)(35,53,71)(36,54,72)(37,55,73)(38,56,74)(39,57,75)(40,58,76)(41,59,77)(42,60,78)(43,61,79)(44,62,80)(45,63,81), (1,58,31)(2,50,77)(3,42,69)(4,34,61)(5,80,53)(6,72,45)(7,64,37)(8,56,29)(9,48,75)(10,40,67)(11,32,59)(12,78,51)(13,70,43)(14,62,35)(15,54,81)(16,46,73)(17,38,65)(18,30,57)(19,76,49)(20,68,41)(21,60,33)(22,52,79)(23,44,71)(24,36,63)(25,28,55)(26,74,47)(27,66,39), (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([[(1,10,19),(2,11,20),(3,12,21),(4,13,22),(5,14,23),(6,15,24),(7,16,25),(8,17,26),(9,18,27),(28,46,64),(30,48,66),(32,50,68),(34,52,70),(36,54,72),(38,56,74),(40,58,76),(42,60,78),(44,62,80)], [(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),(28,46,64),(29,47,65),(30,48,66),(31,49,67),(32,50,68),(33,51,69),(34,52,70),(35,53,71),(36,54,72),(37,55,73),(38,56,74),(39,57,75),(40,58,76),(41,59,77),(42,60,78),(43,61,79),(44,62,80),(45,63,81)], [(1,58,31),(2,50,77),(3,42,69),(4,34,61),(5,80,53),(6,72,45),(7,64,37),(8,56,29),(9,48,75),(10,40,67),(11,32,59),(12,78,51),(13,70,43),(14,62,35),(15,54,81),(16,46,73),(17,38,65),(18,30,57),(19,76,49),(20,68,41),(21,60,33),(22,52,79),(23,44,71),(24,36,63),(25,28,55),(26,74,47),(27,66,39)], [(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 | C3xS3 | S3xC9 | He3.C18 | C32:C6 | C32:C18 |
| kernel | He3.C18 | C9.5He3 | He3.4C6 | C9oHe3 | He3:C2 | He3 | C3xC27 | C3xC9 | C32 | C1 | C9 | C3 |
| # reps | 1 | 1 | 2 | 2 | 6 | 6 | 1 | 2 | 6 | 36 | 1 | 2 |
Matrix representation of He3.C18 ►in GL3(F109) generated by
| 63 | 0 | 46 |
| 0 | 1 | 45 |
| 0 | 0 | 45 |
| 45 | 0 | 0 |
| 0 | 45 | 0 |
| 0 | 0 | 45 |
| 108 | 1 | 0 |
| 108 | 0 | 0 |
| 62 | 0 | 1 |
| 26 | 106 | 83 |
| 0 | 106 | 0 |
| 0 | 77 | 3 |
G:=sub<GL(3,GF(109))| [63,0,0,0,1,0,46,45,45],[45,0,0,0,45,0,0,0,45],[108,108,62,1,0,0,0,0,1],[26,0,0,106,106,77,83,0,3] >;
He3.C18 in GAP, Magma, Sage, TeX
{\rm He}_3.C_{18} % in TeX
G:=Group("He3.C18"); // GroupNames label
G:=SmallGroup(486,26);
// by ID
G=gap.SmallGroup(486,26);
# by ID
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,43,500,867,873,8104,382]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=c^3=1,d^18=b,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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