p-group, metabelian, nilpotent (class 3), monomial
Aliases: He3⋊C9, C32.21He3, C33.21C32, C32.13- 1+2, C3.2C3≀C3, C32⋊C9⋊5C3, (C32×C9)⋊1C3, (C3×He3).3C3, C32.1(C3×C9), C3.7(C32⋊C9), C3.2(He3.C3), C3.2(He3⋊C3), SmallGroup(243,17)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for He3⋊C9
G = < a,b,c,d | a3=b3=c3=d9=1, ab=ba, cac-1=dad-1=ab-1, bc=cb, bd=db, dcd-1=a-1b-1c >
3C3
3C3
3C3
9C3
9C3
9C3
3C32
3C9
3C32
3C32
3C32
3C9
3C9
3C32
3C32
9C9
9C32
9C32
9C9
9C32
3He3
3C33
3He3
Smallest permutation representation of He3⋊C9
►On 81 points
Generators in S81
(1 45 11)(2 71 34)(3 62 46)(4 39 14)(5 65 28)(6 56 49)(7 42 17)(8 68 31)(9 59 52)(10 79 69)(12 27 61)(13 73 72)(15 21 55)(16 76 66)(18 24 58)(19 38 35)(20 64 47)(22 41 29)(23 67 50)(25 44 32)(26 70 53)(30 77 57)(33 80 60)(36 74 63)(37 54 81)(40 48 75)(43 51 78)
(1 80 26)(2 81 27)(3 73 19)(4 74 20)(5 75 21)(6 76 22)(7 77 23)(8 78 24)(9 79 25)(10 32 52)(11 33 53)(12 34 54)(13 35 46)(14 36 47)(15 28 48)(16 29 49)(17 30 50)(18 31 51)(37 61 71)(38 62 72)(39 63 64)(40 55 65)(41 56 66)(42 57 67)(43 58 68)(44 59 69)(45 60 70)
(1 4 7)(2 28 58)(3 41 32)(5 31 61)(6 44 35)(8 34 55)(9 38 29)(10 19 66)(11 47 30)(12 40 24)(13 22 69)(14 50 33)(15 43 27)(16 25 72)(17 53 36)(18 37 21)(20 23 26)(39 57 70)(42 60 64)(45 63 67)(46 76 59)(48 68 81)(49 79 62)(51 71 75)(52 73 56)(54 65 78)(74 77 80)
(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)| (1,45,11)(2,71,34)(3,62,46)(4,39,14)(5,65,28)(6,56,49)(7,42,17)(8,68,31)(9,59,52)(10,79,69)(12,27,61)(13,73,72)(15,21,55)(16,76,66)(18,24,58)(19,38,35)(20,64,47)(22,41,29)(23,67,50)(25,44,32)(26,70,53)(30,77,57)(33,80,60)(36,74,63)(37,54,81)(40,48,75)(43,51,78), (1,80,26)(2,81,27)(3,73,19)(4,74,20)(5,75,21)(6,76,22)(7,77,23)(8,78,24)(9,79,25)(10,32,52)(11,33,53)(12,34,54)(13,35,46)(14,36,47)(15,28,48)(16,29,49)(17,30,50)(18,31,51)(37,61,71)(38,62,72)(39,63,64)(40,55,65)(41,56,66)(42,57,67)(43,58,68)(44,59,69)(45,60,70), (1,4,7)(2,28,58)(3,41,32)(5,31,61)(6,44,35)(8,34,55)(9,38,29)(10,19,66)(11,47,30)(12,40,24)(13,22,69)(14,50,33)(15,43,27)(16,25,72)(17,53,36)(18,37,21)(20,23,26)(39,57,70)(42,60,64)(45,63,67)(46,76,59)(48,68,81)(49,79,62)(51,71,75)(52,73,56)(54,65,78)(74,77,80), (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( (1,45,11)(2,71,34)(3,62,46)(4,39,14)(5,65,28)(6,56,49)(7,42,17)(8,68,31)(9,59,52)(10,79,69)(12,27,61)(13,73,72)(15,21,55)(16,76,66)(18,24,58)(19,38,35)(20,64,47)(22,41,29)(23,67,50)(25,44,32)(26,70,53)(30,77,57)(33,80,60)(36,74,63)(37,54,81)(40,48,75)(43,51,78), (1,80,26)(2,81,27)(3,73,19)(4,74,20)(5,75,21)(6,76,22)(7,77,23)(8,78,24)(9,79,25)(10,32,52)(11,33,53)(12,34,54)(13,35,46)(14,36,47)(15,28,48)(16,29,49)(17,30,50)(18,31,51)(37,61,71)(38,62,72)(39,63,64)(40,55,65)(41,56,66)(42,57,67)(43,58,68)(44,59,69)(45,60,70), (1,4,7)(2,28,58)(3,41,32)(5,31,61)(6,44,35)(8,34,55)(9,38,29)(10,19,66)(11,47,30)(12,40,24)(13,22,69)(14,50,33)(15,43,27)(16,25,72)(17,53,36)(18,37,21)(20,23,26)(39,57,70)(42,60,64)(45,63,67)(46,76,59)(48,68,81)(49,79,62)(51,71,75)(52,73,56)(54,65,78)(74,77,80), (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([[(1,45,11),(2,71,34),(3,62,46),(4,39,14),(5,65,28),(6,56,49),(7,42,17),(8,68,31),(9,59,52),(10,79,69),(12,27,61),(13,73,72),(15,21,55),(16,76,66),(18,24,58),(19,38,35),(20,64,47),(22,41,29),(23,67,50),(25,44,32),(26,70,53),(30,77,57),(33,80,60),(36,74,63),(37,54,81),(40,48,75),(43,51,78)], [(1,80,26),(2,81,27),(3,73,19),(4,74,20),(5,75,21),(6,76,22),(7,77,23),(8,78,24),(9,79,25),(10,32,52),(11,33,53),(12,34,54),(13,35,46),(14,36,47),(15,28,48),(16,29,49),(17,30,50),(18,31,51),(37,61,71),(38,62,72),(39,63,64),(40,55,65),(41,56,66),(42,57,67),(43,58,68),(44,59,69),(45,60,70)], [(1,4,7),(2,28,58),(3,41,32),(5,31,61),(6,44,35),(8,34,55),(9,38,29),(10,19,66),(11,47,30),(12,40,24),(13,22,69),(14,50,33),(15,43,27),(16,25,72),(17,53,36),(18,37,21),(20,23,26),(39,57,70),(42,60,64),(45,63,67),(46,76,59),(48,68,81),(49,79,62),(51,71,75),(52,73,56),(54,65,78),(74,77,80)], [(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)]])
He3⋊C9 is a maximal subgroup of
He3⋊C18 He3⋊D9 He3⋊2D9
51 conjugacy classes
| class | 1 | 3A | ··· | 3H | 3I | ··· | 3N | 3O | ··· | 3T | 9A | ··· | 9R | 9S | ··· | 9AD |
| order | 1 | 3 | ··· | 3 | 3 | ··· | 3 | 3 | ··· | 3 | 9 | ··· | 9 | 9 | ··· | 9 |
| size | 1 | 1 | ··· | 1 | 3 | ··· | 3 | 9 | ··· | 9 | 3 | ··· | 3 | 9 | ··· | 9 |
51 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 3 |
| type | + | |||||||||
| image | C1 | C3 | C3 | C3 | C9 | He3 | 3- 1+2 | C3≀C3 | He3.C3 | He3⋊C3 |
| kernel | He3⋊C9 | C32⋊C9 | C32×C9 | C3×He3 | He3 | C32 | C32 | C3 | C3 | C3 |
| # reps | 1 | 4 | 2 | 2 | 18 | 2 | 4 | 6 | 6 | 6 |
Matrix representation of He3⋊C9 ►in GL4(𝔽19) generated by
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 11 | 0 | 0 |
| 0 | 0 | 11 | 0 |
| 0 | 0 | 0 | 11 |
| 7 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 7 | 0 |
| 0 | 0 | 0 | 11 |
| 17 | 0 | 0 | 0 |
| 0 | 13 | 15 | 15 |
| 0 | 10 | 15 | 10 |
| 0 | 13 | 13 | 10 |
G:=sub<GL(4,GF(19))| [1,0,0,0,0,0,0,1,0,1,0,0,0,0,1,0],[1,0,0,0,0,11,0,0,0,0,11,0,0,0,0,11],[7,0,0,0,0,1,0,0,0,0,7,0,0,0,0,11],[17,0,0,0,0,13,10,13,0,15,15,13,0,15,10,10] >;
He3⋊C9 in GAP, Magma, Sage, TeX
{\rm He}_3\rtimes C_9 % in TeX
G:=Group("He3:C9"); // GroupNames label
G:=SmallGroup(243,17);
// by ID
G=gap.SmallGroup(243,17);
# by ID
G:=PCGroup([5,-3,3,-3,3,-3,135,121,1352,457]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=c^3=d^9=1,a*b=b*a,c*a*c^-1=d*a*d^-1=a*b^-1,b*c=c*b,b*d=d*b,d*c*d^-1=a^-1*b^-1*c>;
// generators/relations
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