direct product, p-group, metabelian, nilpotent (class 3), monomial
Aliases: C3×He3⋊C3, He3⋊2C32, C32.11He3, C32.3C33, C33.33C32, (C3×He3)⋊4C3, (C3×C9)⋊7C32, (C32×C9)⋊6C3, C3.9(C3×He3), SmallGroup(243,53)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C3×He3⋊C3
G = < a,b,c,d,e | a3=b3=c3=d3=e3=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=ebe-1=bc-1, cd=dc, ce=ec, ede-1=bcd >
Subgroups: 288 in 84 conjugacy classes, 36 normal (7 characteristic)
C1, C3, C3, C3, C9, C32, C32, C32, C3×C9, C3×C9, He3, He3, C33, C33, He3⋊C3, C32×C9, C3×He3, C3×He3⋊C3
Quotients: C1, C3, C32, He3, C33, He3⋊C3, C3×He3, C3×He3⋊C3
►On 81 points
Generators in S81
(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)
(1 17 9)(2 18 7)(3 16 8)(4 23 14)(5 24 15)(6 22 13)(10 79 21)(11 80 19)(12 81 20)(25 38 30)(26 39 28)(27 37 29)(31 47 42)(32 48 40)(33 46 41)(34 50 45)(35 51 43)(36 49 44)(52 65 57)(53 66 55)(54 64 56)(58 74 69)(59 75 67)(60 73 68)(61 77 72)(62 78 70)(63 76 71)
(1 4 26)(2 5 27)(3 6 25)(7 15 29)(8 13 30)(9 14 28)(10 62 58)(11 63 59)(12 61 60)(16 22 38)(17 23 39)(18 24 37)(19 71 67)(20 72 68)(21 70 69)(31 57 35)(32 55 36)(33 56 34)(40 66 44)(41 64 45)(42 65 43)(46 54 50)(47 52 51)(48 53 49)(73 81 77)(74 79 78)(75 80 76)
(1 46 10)(2 47 11)(3 48 12)(4 54 62)(5 52 63)(6 53 61)(7 35 71)(8 36 72)(9 34 70)(13 32 68)(14 33 69)(15 31 67)(16 66 73)(17 64 74)(18 65 75)(19 29 57)(20 30 55)(21 28 56)(22 44 81)(23 45 79)(24 43 80)(25 49 60)(26 50 58)(27 51 59)(37 42 76)(38 40 77)(39 41 78)
(7 29 15)(8 30 13)(9 28 14)(10 74 69)(11 75 67)(12 73 68)(16 22 38)(17 23 39)(18 24 37)(19 63 80)(20 61 81)(21 62 79)(31 43 47)(32 44 48)(33 45 46)(34 64 50)(35 65 51)(36 66 49)(40 53 55)(41 54 56)(42 52 57)(58 78 70)(59 76 71)(60 77 72)
G:=sub<Sym(81)| (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), (1,17,9)(2,18,7)(3,16,8)(4,23,14)(5,24,15)(6,22,13)(10,79,21)(11,80,19)(12,81,20)(25,38,30)(26,39,28)(27,37,29)(31,47,42)(32,48,40)(33,46,41)(34,50,45)(35,51,43)(36,49,44)(52,65,57)(53,66,55)(54,64,56)(58,74,69)(59,75,67)(60,73,68)(61,77,72)(62,78,70)(63,76,71), (1,4,26)(2,5,27)(3,6,25)(7,15,29)(8,13,30)(9,14,28)(10,62,58)(11,63,59)(12,61,60)(16,22,38)(17,23,39)(18,24,37)(19,71,67)(20,72,68)(21,70,69)(31,57,35)(32,55,36)(33,56,34)(40,66,44)(41,64,45)(42,65,43)(46,54,50)(47,52,51)(48,53,49)(73,81,77)(74,79,78)(75,80,76), (1,46,10)(2,47,11)(3,48,12)(4,54,62)(5,52,63)(6,53,61)(7,35,71)(8,36,72)(9,34,70)(13,32,68)(14,33,69)(15,31,67)(16,66,73)(17,64,74)(18,65,75)(19,29,57)(20,30,55)(21,28,56)(22,44,81)(23,45,79)(24,43,80)(25,49,60)(26,50,58)(27,51,59)(37,42,76)(38,40,77)(39,41,78), (7,29,15)(8,30,13)(9,28,14)(10,74,69)(11,75,67)(12,73,68)(16,22,38)(17,23,39)(18,24,37)(19,63,80)(20,61,81)(21,62,79)(31,43,47)(32,44,48)(33,45,46)(34,64,50)(35,65,51)(36,66,49)(40,53,55)(41,54,56)(42,52,57)(58,78,70)(59,76,71)(60,77,72)>;
G:=Group( (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), (1,17,9)(2,18,7)(3,16,8)(4,23,14)(5,24,15)(6,22,13)(10,79,21)(11,80,19)(12,81,20)(25,38,30)(26,39,28)(27,37,29)(31,47,42)(32,48,40)(33,46,41)(34,50,45)(35,51,43)(36,49,44)(52,65,57)(53,66,55)(54,64,56)(58,74,69)(59,75,67)(60,73,68)(61,77,72)(62,78,70)(63,76,71), (1,4,26)(2,5,27)(3,6,25)(7,15,29)(8,13,30)(9,14,28)(10,62,58)(11,63,59)(12,61,60)(16,22,38)(17,23,39)(18,24,37)(19,71,67)(20,72,68)(21,70,69)(31,57,35)(32,55,36)(33,56,34)(40,66,44)(41,64,45)(42,65,43)(46,54,50)(47,52,51)(48,53,49)(73,81,77)(74,79,78)(75,80,76), (1,46,10)(2,47,11)(3,48,12)(4,54,62)(5,52,63)(6,53,61)(7,35,71)(8,36,72)(9,34,70)(13,32,68)(14,33,69)(15,31,67)(16,66,73)(17,64,74)(18,65,75)(19,29,57)(20,30,55)(21,28,56)(22,44,81)(23,45,79)(24,43,80)(25,49,60)(26,50,58)(27,51,59)(37,42,76)(38,40,77)(39,41,78), (7,29,15)(8,30,13)(9,28,14)(10,74,69)(11,75,67)(12,73,68)(16,22,38)(17,23,39)(18,24,37)(19,63,80)(20,61,81)(21,62,79)(31,43,47)(32,44,48)(33,45,46)(34,64,50)(35,65,51)(36,66,49)(40,53,55)(41,54,56)(42,52,57)(58,78,70)(59,76,71)(60,77,72) );
G=PermutationGroup([[(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)], [(1,17,9),(2,18,7),(3,16,8),(4,23,14),(5,24,15),(6,22,13),(10,79,21),(11,80,19),(12,81,20),(25,38,30),(26,39,28),(27,37,29),(31,47,42),(32,48,40),(33,46,41),(34,50,45),(35,51,43),(36,49,44),(52,65,57),(53,66,55),(54,64,56),(58,74,69),(59,75,67),(60,73,68),(61,77,72),(62,78,70),(63,76,71)], [(1,4,26),(2,5,27),(3,6,25),(7,15,29),(8,13,30),(9,14,28),(10,62,58),(11,63,59),(12,61,60),(16,22,38),(17,23,39),(18,24,37),(19,71,67),(20,72,68),(21,70,69),(31,57,35),(32,55,36),(33,56,34),(40,66,44),(41,64,45),(42,65,43),(46,54,50),(47,52,51),(48,53,49),(73,81,77),(74,79,78),(75,80,76)], [(1,46,10),(2,47,11),(3,48,12),(4,54,62),(5,52,63),(6,53,61),(7,35,71),(8,36,72),(9,34,70),(13,32,68),(14,33,69),(15,31,67),(16,66,73),(17,64,74),(18,65,75),(19,29,57),(20,30,55),(21,28,56),(22,44,81),(23,45,79),(24,43,80),(25,49,60),(26,50,58),(27,51,59),(37,42,76),(38,40,77),(39,41,78)], [(7,29,15),(8,30,13),(9,28,14),(10,74,69),(11,75,67),(12,73,68),(16,22,38),(17,23,39),(18,24,37),(19,63,80),(20,61,81),(21,62,79),(31,43,47),(32,44,48),(33,45,46),(34,64,50),(35,65,51),(36,66,49),(40,53,55),(41,54,56),(42,52,57),(58,78,70),(59,76,71),(60,77,72)]])
C3×He3⋊C3 is a maximal subgroup of
He3⋊C3⋊2S3 He3⋊C3⋊3S3 C3⋊(He3⋊S3)
51 conjugacy classes
| class | 1 | 3A | ··· | 3H | 3I | ··· | 3N | 3O | ··· | 3AF | 9A | ··· | 9R |
| order | 1 | 3 | ··· | 3 | 3 | ··· | 3 | 3 | ··· | 3 | 9 | ··· | 9 |
| size | 1 | 1 | ··· | 1 | 3 | ··· | 3 | 9 | ··· | 9 | 3 | ··· | 3 |
51 irreducible representations
| dim | 1 | 1 | 1 | 1 | 3 | 3 |
| type | + | |||||
| image | C1 | C3 | C3 | C3 | He3 | He3⋊C3 |
| kernel | C3×He3⋊C3 | He3⋊C3 | C32×C9 | C3×He3 | C32 | C3 |
| # reps | 1 | 18 | 2 | 6 | 6 | 18 |
Matrix representation of C3×He3⋊C3 ►in GL4(𝔽19) generated by
| 7 | 0 | 0 | 0 |
| 0 | 7 | 0 | 0 |
| 0 | 0 | 7 | 0 |
| 0 | 0 | 0 | 7 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 6 | 18 | 18 |
| 1 | 0 | 0 | 0 |
| 0 | 11 | 0 | 0 |
| 0 | 0 | 11 | 0 |
| 0 | 0 | 0 | 11 |
| 7 | 0 | 0 | 0 |
| 0 | 17 | 17 | 3 |
| 0 | 3 | 5 | 0 |
| 0 | 2 | 0 | 16 |
| 7 | 0 | 0 | 0 |
| 0 | 1 | 1 | 8 |
| 0 | 0 | 7 | 0 |
| 0 | 0 | 0 | 11 |
G:=sub<GL(4,GF(19))| [7,0,0,0,0,7,0,0,0,0,7,0,0,0,0,7],[1,0,0,0,0,1,0,6,0,0,0,18,0,0,1,18],[1,0,0,0,0,11,0,0,0,0,11,0,0,0,0,11],[7,0,0,0,0,17,3,2,0,17,5,0,0,3,0,16],[7,0,0,0,0,1,0,0,0,1,7,0,0,8,0,11] >;
C3×He3⋊C3 in GAP, Magma, Sage, TeX
C_3\times {\rm He}_3\rtimes C_3 % in TeX
G:=Group("C3xHe3:C3"); // GroupNames label
G:=SmallGroup(243,53);
// by ID
G=gap.SmallGroup(243,53);
# by ID
G:=PCGroup([5,-3,3,3,-3,-3,301,546,2163]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^3=e^3=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,d*b*d^-1=e*b*e^-1=b*c^-1,c*d=d*c,c*e=e*c,e*d*e^-1=b*c*d>;
// generators/relations