non-abelian, supersoluble, monomial
Aliases: He3.5C18, (C3×C27)⋊8S3, C27○He3⋊2C2, C27.2(C3⋊S3), C9○He3.5C6, C32.5(S3×C9), He3⋊C2.3C9, He3.4C6.3C3, C9.7(C3×C3⋊S3), C3.7(C9×C3⋊S3), (C3×C9).26(C3×S3), SmallGroup(486,164)
Series: Derived ►Chief ►Lower central ►Upper central
| He3 — He3.5C18 |
Generators and relations for He3.5C18
G = < a,b,c,d | a3=b3=c3=1, d18=b, ab=ba, cac-1=ab-1, dad-1=a-1, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 150 in 62 conjugacy classes, 22 normal (10 characteristic)
C1, C2, C3, C3, S3, C6, C9, C9, C32, C18, C3×S3, C27, C27, C3×C9, He3, 3- 1+2, C54, S3×C9, He3⋊C2, C3×C27, C27⋊C3, C9○He3, S3×C27, He3.4C6, C27○He3, He3.5C18
Quotients: C1, C2, C3, S3, C6, C9, C18, C3×S3, C3⋊S3, S3×C9, C3×C3⋊S3, C9×C3⋊S3, He3.5C18
►On 81 points
Generators in S81
(1 64 37)(2 38 65)(3 66 39)(4 40 67)(5 68 41)(6 42 69)(7 70 43)(8 44 71)(9 72 45)(10 46 73)(11 74 47)(12 48 75)(13 76 49)(14 50 77)(15 78 51)(16 52 79)(17 80 53)(18 54 81)(19 28 55)(20 56 29)(21 30 57)(22 58 31)(23 32 59)(24 60 33)(25 34 61)(26 62 35)(27 36 63)
(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)
(28 46 64)(29 65 47)(30 48 66)(31 67 49)(32 50 68)(33 69 51)(34 52 70)(35 71 53)(36 54 72)(37 73 55)(38 56 74)(39 75 57)(40 58 76)(41 77 59)(42 60 78)(43 79 61)(44 62 80)(45 81 63)
(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)| (1,64,37)(2,38,65)(3,66,39)(4,40,67)(5,68,41)(6,42,69)(7,70,43)(8,44,71)(9,72,45)(10,46,73)(11,74,47)(12,48,75)(13,76,49)(14,50,77)(15,78,51)(16,52,79)(17,80,53)(18,54,81)(19,28,55)(20,56,29)(21,30,57)(22,58,31)(23,32,59)(24,60,33)(25,34,61)(26,62,35)(27,36,63), (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), (28,46,64)(29,65,47)(30,48,66)(31,67,49)(32,50,68)(33,69,51)(34,52,70)(35,71,53)(36,54,72)(37,73,55)(38,56,74)(39,75,57)(40,58,76)(41,77,59)(42,60,78)(43,79,61)(44,62,80)(45,81,63), (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( (1,64,37)(2,38,65)(3,66,39)(4,40,67)(5,68,41)(6,42,69)(7,70,43)(8,44,71)(9,72,45)(10,46,73)(11,74,47)(12,48,75)(13,76,49)(14,50,77)(15,78,51)(16,52,79)(17,80,53)(18,54,81)(19,28,55)(20,56,29)(21,30,57)(22,58,31)(23,32,59)(24,60,33)(25,34,61)(26,62,35)(27,36,63), (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), (28,46,64)(29,65,47)(30,48,66)(31,67,49)(32,50,68)(33,69,51)(34,52,70)(35,71,53)(36,54,72)(37,73,55)(38,56,74)(39,75,57)(40,58,76)(41,77,59)(42,60,78)(43,79,61)(44,62,80)(45,81,63), (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([[(1,64,37),(2,38,65),(3,66,39),(4,40,67),(5,68,41),(6,42,69),(7,70,43),(8,44,71),(9,72,45),(10,46,73),(11,74,47),(12,48,75),(13,76,49),(14,50,77),(15,78,51),(16,52,79),(17,80,53),(18,54,81),(19,28,55),(20,56,29),(21,30,57),(22,58,31),(23,32,59),(24,60,33),(25,34,61),(26,62,35),(27,36,63)], [(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)], [(28,46,64),(29,65,47),(30,48,66),(31,67,49),(32,50,68),(33,69,51),(34,52,70),(35,71,53),(36,54,72),(37,73,55),(38,56,74),(39,75,57),(40,58,76),(41,77,59),(42,60,78),(43,79,61),(44,62,80),(45,81,63)], [(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)]])
90 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 3F | 6A | 6B | 9A | ··· | 9F | 9G | ··· | 9N | 18A | ··· | 18F | 27A | ··· | 27R | 27S | ··· | 27AP | 54A | ··· | 54R |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 6 | 6 | 9 | ··· | 9 | 9 | ··· | 9 | 18 | ··· | 18 | 27 | ··· | 27 | 27 | ··· | 27 | 54 | ··· | 54 |
| size | 1 | 9 | 1 | 1 | 6 | 6 | 6 | 6 | 9 | 9 | 1 | ··· | 1 | 6 | ··· | 6 | 9 | ··· | 9 | 1 | ··· | 1 | 6 | ··· | 6 | 9 | ··· | 9 |
90 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 3 |
| type | + | + | + | |||||||
| image | C1 | C2 | C3 | C6 | C9 | C18 | S3 | C3×S3 | S3×C9 | He3.5C18 |
| kernel | He3.5C18 | C27○He3 | He3.4C6 | C9○He3 | He3⋊C2 | He3 | C3×C27 | C3×C9 | C32 | C1 |
| # reps | 1 | 1 | 2 | 2 | 6 | 6 | 4 | 8 | 24 | 36 |
Matrix representation of He3.5C18 ►in GL3(𝔽109) generated by
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 45 | 0 | 0 |
| 0 | 45 | 0 |
| 0 | 0 | 45 |
| 1 | 0 | 0 |
| 0 | 63 | 0 |
| 0 | 0 | 45 |
| 5 | 0 | 0 |
| 0 | 0 | 5 |
| 0 | 5 | 0 |
G:=sub<GL(3,GF(109))| [0,0,1,1,0,0,0,1,0],[45,0,0,0,45,0,0,0,45],[1,0,0,0,63,0,0,0,45],[5,0,0,0,0,5,0,5,0] >;
He3.5C18 in GAP, Magma, Sage, TeX
{\rm He}_3._5C_{18} % in TeX
G:=Group("He3.5C18"); // GroupNames label
G:=SmallGroup(486,164);
// by ID
G=gap.SmallGroup(486,164);
# by ID
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,43,500,867,3244,382]);
// 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*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations