direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: C32×He3, C34⋊3C3, C32⋊C33, C3.1C34, C33⋊5C32, SmallGroup(243,62)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C32×He3
G = < a,b,c,d,e | a3=b3=c3=d3=e3=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=cd-1, de=ed >
Subgroups: 882 in 450 conjugacy classes, 234 normal (4 characteristic)
C1, C3, C3, C3, C32, C32, He3, C33, C33, C33, C3×He3, C34, C32×He3
Quotients: C1, C3, C32, He3, C33, C3×He3, C34, C32×He3
►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 19 11)(2 20 12)(3 21 10)(4 9 18)(5 7 16)(6 8 17)(13 79 27)(14 80 25)(15 81 26)(22 44 33)(23 45 31)(24 43 32)(28 50 42)(29 51 40)(30 49 41)(34 47 39)(35 48 37)(36 46 38)(52 71 60)(53 72 58)(54 70 59)(55 77 69)(56 78 67)(57 76 68)(61 74 66)(62 75 64)(63 73 65)
(1 13 36)(2 14 34)(3 15 35)(4 55 58)(5 56 59)(6 57 60)(7 78 54)(8 76 52)(9 77 53)(10 26 37)(11 27 38)(12 25 39)(16 67 70)(17 68 71)(18 69 72)(19 79 46)(20 80 47)(21 81 48)(22 73 49)(23 74 50)(24 75 51)(28 31 61)(29 32 62)(30 33 63)(40 43 64)(41 44 65)(42 45 66)
(1 7 45)(2 8 43)(3 9 44)(4 22 10)(5 23 11)(6 24 12)(13 78 66)(14 76 64)(15 77 65)(16 31 19)(17 32 20)(18 33 21)(25 57 75)(26 55 73)(27 56 74)(28 46 70)(29 47 71)(30 48 72)(34 52 40)(35 53 41)(36 54 42)(37 58 49)(38 59 50)(39 60 51)(61 79 67)(62 80 68)(63 81 69)
(1 79 59)(2 80 60)(3 81 58)(4 77 30)(5 78 28)(6 76 29)(7 67 50)(8 68 51)(9 69 49)(10 15 72)(11 13 70)(12 14 71)(16 56 42)(17 57 40)(18 55 41)(19 27 54)(20 25 52)(21 26 53)(22 65 48)(23 66 46)(24 64 47)(31 74 36)(32 75 34)(33 73 35)(37 44 63)(38 45 61)(39 43 62)
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), (1,19,11)(2,20,12)(3,21,10)(4,9,18)(5,7,16)(6,8,17)(13,79,27)(14,80,25)(15,81,26)(22,44,33)(23,45,31)(24,43,32)(28,50,42)(29,51,40)(30,49,41)(34,47,39)(35,48,37)(36,46,38)(52,71,60)(53,72,58)(54,70,59)(55,77,69)(56,78,67)(57,76,68)(61,74,66)(62,75,64)(63,73,65), (1,13,36)(2,14,34)(3,15,35)(4,55,58)(5,56,59)(6,57,60)(7,78,54)(8,76,52)(9,77,53)(10,26,37)(11,27,38)(12,25,39)(16,67,70)(17,68,71)(18,69,72)(19,79,46)(20,80,47)(21,81,48)(22,73,49)(23,74,50)(24,75,51)(28,31,61)(29,32,62)(30,33,63)(40,43,64)(41,44,65)(42,45,66), (1,7,45)(2,8,43)(3,9,44)(4,22,10)(5,23,11)(6,24,12)(13,78,66)(14,76,64)(15,77,65)(16,31,19)(17,32,20)(18,33,21)(25,57,75)(26,55,73)(27,56,74)(28,46,70)(29,47,71)(30,48,72)(34,52,40)(35,53,41)(36,54,42)(37,58,49)(38,59,50)(39,60,51)(61,79,67)(62,80,68)(63,81,69), (1,79,59)(2,80,60)(3,81,58)(4,77,30)(5,78,28)(6,76,29)(7,67,50)(8,68,51)(9,69,49)(10,15,72)(11,13,70)(12,14,71)(16,56,42)(17,57,40)(18,55,41)(19,27,54)(20,25,52)(21,26,53)(22,65,48)(23,66,46)(24,64,47)(31,74,36)(32,75,34)(33,73,35)(37,44,63)(38,45,61)(39,43,62)>;
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), (1,19,11)(2,20,12)(3,21,10)(4,9,18)(5,7,16)(6,8,17)(13,79,27)(14,80,25)(15,81,26)(22,44,33)(23,45,31)(24,43,32)(28,50,42)(29,51,40)(30,49,41)(34,47,39)(35,48,37)(36,46,38)(52,71,60)(53,72,58)(54,70,59)(55,77,69)(56,78,67)(57,76,68)(61,74,66)(62,75,64)(63,73,65), (1,13,36)(2,14,34)(3,15,35)(4,55,58)(5,56,59)(6,57,60)(7,78,54)(8,76,52)(9,77,53)(10,26,37)(11,27,38)(12,25,39)(16,67,70)(17,68,71)(18,69,72)(19,79,46)(20,80,47)(21,81,48)(22,73,49)(23,74,50)(24,75,51)(28,31,61)(29,32,62)(30,33,63)(40,43,64)(41,44,65)(42,45,66), (1,7,45)(2,8,43)(3,9,44)(4,22,10)(5,23,11)(6,24,12)(13,78,66)(14,76,64)(15,77,65)(16,31,19)(17,32,20)(18,33,21)(25,57,75)(26,55,73)(27,56,74)(28,46,70)(29,47,71)(30,48,72)(34,52,40)(35,53,41)(36,54,42)(37,58,49)(38,59,50)(39,60,51)(61,79,67)(62,80,68)(63,81,69), (1,79,59)(2,80,60)(3,81,58)(4,77,30)(5,78,28)(6,76,29)(7,67,50)(8,68,51)(9,69,49)(10,15,72)(11,13,70)(12,14,71)(16,56,42)(17,57,40)(18,55,41)(19,27,54)(20,25,52)(21,26,53)(22,65,48)(23,66,46)(24,64,47)(31,74,36)(32,75,34)(33,73,35)(37,44,63)(38,45,61)(39,43,62) );
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)], [(1,19,11),(2,20,12),(3,21,10),(4,9,18),(5,7,16),(6,8,17),(13,79,27),(14,80,25),(15,81,26),(22,44,33),(23,45,31),(24,43,32),(28,50,42),(29,51,40),(30,49,41),(34,47,39),(35,48,37),(36,46,38),(52,71,60),(53,72,58),(54,70,59),(55,77,69),(56,78,67),(57,76,68),(61,74,66),(62,75,64),(63,73,65)], [(1,13,36),(2,14,34),(3,15,35),(4,55,58),(5,56,59),(6,57,60),(7,78,54),(8,76,52),(9,77,53),(10,26,37),(11,27,38),(12,25,39),(16,67,70),(17,68,71),(18,69,72),(19,79,46),(20,80,47),(21,81,48),(22,73,49),(23,74,50),(24,75,51),(28,31,61),(29,32,62),(30,33,63),(40,43,64),(41,44,65),(42,45,66)], [(1,7,45),(2,8,43),(3,9,44),(4,22,10),(5,23,11),(6,24,12),(13,78,66),(14,76,64),(15,77,65),(16,31,19),(17,32,20),(18,33,21),(25,57,75),(26,55,73),(27,56,74),(28,46,70),(29,47,71),(30,48,72),(34,52,40),(35,53,41),(36,54,42),(37,58,49),(38,59,50),(39,60,51),(61,79,67),(62,80,68),(63,81,69)], [(1,79,59),(2,80,60),(3,81,58),(4,77,30),(5,78,28),(6,76,29),(7,67,50),(8,68,51),(9,69,49),(10,15,72),(11,13,70),(12,14,71),(16,56,42),(17,57,40),(18,55,41),(19,27,54),(20,25,52),(21,26,53),(22,65,48),(23,66,46),(24,64,47),(31,74,36),(32,75,34),(33,73,35),(37,44,63),(38,45,61),(39,43,62)]])
C32×He3 is a maximal subgroup of
C34⋊10C6 C34⋊13S3
99 conjugacy classes
| class | 1 | 3A | ··· | 3Z | 3AA | ··· | 3CT |
| order | 1 | 3 | ··· | 3 | 3 | ··· | 3 |
| size | 1 | 1 | ··· | 1 | 3 | ··· | 3 |
99 irreducible representations
| dim | 1 | 1 | 1 | 3 |
| type | + | |||
| image | C1 | C3 | C3 | He3 |
| kernel | C32×He3 | C3×He3 | C34 | C32 |
| # reps | 1 | 72 | 8 | 18 |
Matrix representation of C32×He3 ►in GL5(𝔽7)
| 4 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 2 | 0 | 0 |
| 0 | 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 2 |
| 2 | 0 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 4 |
| 4 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 5 | 6 | 5 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 4 |
| 4 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 3 | 5 | 3 |
| 0 | 0 | 5 | 4 | 2 |
G:=sub<GL(5,GF(7))| [4,0,0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,0,2,0,0,0,0,0,2],[2,0,0,0,0,0,2,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4],[4,0,0,0,0,0,1,0,0,0,0,0,0,5,0,0,0,4,6,0,0,0,0,5,1],[1,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4],[4,0,0,0,0,0,1,0,0,0,0,0,0,3,5,0,0,4,5,4,0,0,0,3,2] >;
C32×He3 in GAP, Magma, Sage, TeX
C_3^2\times {\rm He}_3 % in TeX
G:=Group("C3^2xHe3"); // GroupNames label
G:=SmallGroup(243,62);
// by ID
G=gap.SmallGroup(243,62);
# by ID
G:=PCGroup([5,-3,3,3,3,-3,841]);
// 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,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=c*d^-1,d*e=e*d>;
// generators/relations