p-group, metabelian, nilpotent (class 3), monomial
Aliases: C32.24He3, C33.1C32, C3.4C3≀C3, C32⋊C9⋊1C3, (C3×He3)⋊1C3, C3.3(He3⋊C3), SmallGroup(243,3)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C32.24He3
G = < a,b,c,d,e | a3=b3=c3=d3=e3=1, ab=ba, ac=ca, ad=da, ae=ea, dcd-1=bc=cb, bd=db, be=eb, ece-1=acd-1, ede-1=a-1d >
9C3
9C3
9C3
9C3
9C3
9C3
9C3
3C32
3C32
3C32
3C32
3C32
3C32
3C32
3C32
3C32
3C32
9C32
9C9
9C32
9C32
9C9
9C32
9C32
9C32
3He3
3C33
3C33
3He3
3He3
3He3
3He3
3He3
Smallest permutation representation of C32.24He3
►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 14 10)(2 15 11)(3 13 12)(4 81 8)(5 79 9)(6 80 7)(16 23 19)(17 24 20)(18 22 21)(25 32 28)(26 33 29)(27 31 30)(34 41 37)(35 42 38)(36 40 39)(43 50 46)(44 51 47)(45 49 48)(52 59 55)(53 60 56)(54 58 57)(61 68 64)(62 69 65)(63 67 66)(70 77 73)(71 78 74)(72 76 75)
(1 3 2)(4 63 72)(5 61 70)(6 62 71)(7 65 74)(8 66 75)(9 64 73)(10 12 11)(13 15 14)(16 21 24)(17 19 22)(18 20 23)(25 31 29)(26 32 30)(27 33 28)(34 54 51)(35 52 49)(36 53 50)(37 57 44)(38 55 45)(39 56 43)(40 60 46)(41 58 47)(42 59 48)(67 76 81)(68 77 79)(69 78 80)
(1 25 16)(2 26 17)(3 27 18)(4 74 64)(5 75 65)(6 73 66)(7 77 67)(8 78 68)(9 76 69)(10 28 19)(11 29 20)(12 30 21)(13 31 22)(14 32 23)(15 33 24)(34 53 45)(35 54 43)(36 52 44)(37 56 48)(38 57 46)(39 55 47)(40 59 51)(41 60 49)(42 58 50)(61 81 71)(62 79 72)(63 80 70)
(1 61 34)(2 62 35)(3 63 36)(4 57 30)(5 55 28)(6 56 29)(7 60 33)(8 58 31)(9 59 32)(10 64 37)(11 65 38)(12 66 39)(13 67 40)(14 68 41)(15 69 42)(16 70 43)(17 71 44)(18 72 45)(19 73 46)(20 74 47)(21 75 48)(22 76 49)(23 77 50)(24 78 51)(25 79 52)(26 80 53)(27 81 54)
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), (1,14,10)(2,15,11)(3,13,12)(4,81,8)(5,79,9)(6,80,7)(16,23,19)(17,24,20)(18,22,21)(25,32,28)(26,33,29)(27,31,30)(34,41,37)(35,42,38)(36,40,39)(43,50,46)(44,51,47)(45,49,48)(52,59,55)(53,60,56)(54,58,57)(61,68,64)(62,69,65)(63,67,66)(70,77,73)(71,78,74)(72,76,75), (1,3,2)(4,63,72)(5,61,70)(6,62,71)(7,65,74)(8,66,75)(9,64,73)(10,12,11)(13,15,14)(16,21,24)(17,19,22)(18,20,23)(25,31,29)(26,32,30)(27,33,28)(34,54,51)(35,52,49)(36,53,50)(37,57,44)(38,55,45)(39,56,43)(40,60,46)(41,58,47)(42,59,48)(67,76,81)(68,77,79)(69,78,80), (1,25,16)(2,26,17)(3,27,18)(4,74,64)(5,75,65)(6,73,66)(7,77,67)(8,78,68)(9,76,69)(10,28,19)(11,29,20)(12,30,21)(13,31,22)(14,32,23)(15,33,24)(34,53,45)(35,54,43)(36,52,44)(37,56,48)(38,57,46)(39,55,47)(40,59,51)(41,60,49)(42,58,50)(61,81,71)(62,79,72)(63,80,70), (1,61,34)(2,62,35)(3,63,36)(4,57,30)(5,55,28)(6,56,29)(7,60,33)(8,58,31)(9,59,32)(10,64,37)(11,65,38)(12,66,39)(13,67,40)(14,68,41)(15,69,42)(16,70,43)(17,71,44)(18,72,45)(19,73,46)(20,74,47)(21,75,48)(22,76,49)(23,77,50)(24,78,51)(25,79,52)(26,80,53)(27,81,54)>;
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), (1,14,10)(2,15,11)(3,13,12)(4,81,8)(5,79,9)(6,80,7)(16,23,19)(17,24,20)(18,22,21)(25,32,28)(26,33,29)(27,31,30)(34,41,37)(35,42,38)(36,40,39)(43,50,46)(44,51,47)(45,49,48)(52,59,55)(53,60,56)(54,58,57)(61,68,64)(62,69,65)(63,67,66)(70,77,73)(71,78,74)(72,76,75), (1,3,2)(4,63,72)(5,61,70)(6,62,71)(7,65,74)(8,66,75)(9,64,73)(10,12,11)(13,15,14)(16,21,24)(17,19,22)(18,20,23)(25,31,29)(26,32,30)(27,33,28)(34,54,51)(35,52,49)(36,53,50)(37,57,44)(38,55,45)(39,56,43)(40,60,46)(41,58,47)(42,59,48)(67,76,81)(68,77,79)(69,78,80), (1,25,16)(2,26,17)(3,27,18)(4,74,64)(5,75,65)(6,73,66)(7,77,67)(8,78,68)(9,76,69)(10,28,19)(11,29,20)(12,30,21)(13,31,22)(14,32,23)(15,33,24)(34,53,45)(35,54,43)(36,52,44)(37,56,48)(38,57,46)(39,55,47)(40,59,51)(41,60,49)(42,58,50)(61,81,71)(62,79,72)(63,80,70), (1,61,34)(2,62,35)(3,63,36)(4,57,30)(5,55,28)(6,56,29)(7,60,33)(8,58,31)(9,59,32)(10,64,37)(11,65,38)(12,66,39)(13,67,40)(14,68,41)(15,69,42)(16,70,43)(17,71,44)(18,72,45)(19,73,46)(20,74,47)(21,75,48)(22,76,49)(23,77,50)(24,78,51)(25,79,52)(26,80,53)(27,81,54) );
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)], [(1,14,10),(2,15,11),(3,13,12),(4,81,8),(5,79,9),(6,80,7),(16,23,19),(17,24,20),(18,22,21),(25,32,28),(26,33,29),(27,31,30),(34,41,37),(35,42,38),(36,40,39),(43,50,46),(44,51,47),(45,49,48),(52,59,55),(53,60,56),(54,58,57),(61,68,64),(62,69,65),(63,67,66),(70,77,73),(71,78,74),(72,76,75)], [(1,3,2),(4,63,72),(5,61,70),(6,62,71),(7,65,74),(8,66,75),(9,64,73),(10,12,11),(13,15,14),(16,21,24),(17,19,22),(18,20,23),(25,31,29),(26,32,30),(27,33,28),(34,54,51),(35,52,49),(36,53,50),(37,57,44),(38,55,45),(39,56,43),(40,60,46),(41,58,47),(42,59,48),(67,76,81),(68,77,79),(69,78,80)], [(1,25,16),(2,26,17),(3,27,18),(4,74,64),(5,75,65),(6,73,66),(7,77,67),(8,78,68),(9,76,69),(10,28,19),(11,29,20),(12,30,21),(13,31,22),(14,32,23),(15,33,24),(34,53,45),(35,54,43),(36,52,44),(37,56,48),(38,57,46),(39,55,47),(40,59,51),(41,60,49),(42,58,50),(61,81,71),(62,79,72),(63,80,70)], [(1,61,34),(2,62,35),(3,63,36),(4,57,30),(5,55,28),(6,56,29),(7,60,33),(8,58,31),(9,59,32),(10,64,37),(11,65,38),(12,66,39),(13,67,40),(14,68,41),(15,69,42),(16,70,43),(17,71,44),(18,72,45),(19,73,46),(20,74,47),(21,75,48),(22,76,49),(23,77,50),(24,78,51),(25,79,52),(26,80,53),(27,81,54)]])
C32.24He3 is a maximal subgroup of
C3.C3≀S3 C32⋊C9.S3 (C3×He3)⋊S3
35 conjugacy classes
| class | 1 | 3A | ··· | 3H | 3I | ··· | 3V | 9A | ··· | 9L |
| order | 1 | 3 | ··· | 3 | 3 | ··· | 3 | 9 | ··· | 9 |
| size | 1 | 1 | ··· | 1 | 9 | ··· | 9 | 9 | ··· | 9 |
35 irreducible representations
| dim | 1 | 1 | 1 | 3 | 3 | 3 |
| type | + | |||||
| image | C1 | C3 | C3 | He3 | C3≀C3 | He3⋊C3 |
| kernel | C32.24He3 | C32⋊C9 | C3×He3 | C32 | C3 | C3 |
| # reps | 1 | 4 | 4 | 2 | 12 | 12 |
Matrix representation of C32.24He3 ►in GL6(𝔽19)
| 11 | 0 | 0 | 0 | 0 | 0 |
| 0 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 0 | 0 | 0 |
| 0 | 0 | 0 | 11 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 0 |
| 0 | 0 | 0 | 0 | 0 | 11 |
| 7 | 0 | 0 | 0 | 0 | 0 |
| 0 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 7 | 0 | 0 | 0 | 0 | 0 |
| 18 | 1 | 0 | 0 | 0 | 0 |
| 7 | 0 | 11 | 0 | 0 | 0 |
| 0 | 0 | 0 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 0 |
| 0 | 0 | 0 | 11 | 16 | 1 |
| 1 | 6 | 0 | 0 | 0 | 0 |
| 0 | 18 | 1 | 0 | 0 | 0 |
| 0 | 18 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 11 | 0 |
| 0 | 0 | 0 | 0 | 11 | 0 |
| 0 | 0 | 0 | 8 | 15 | 7 |
| 9 | 4 | 6 | 0 | 0 | 0 |
| 9 | 0 | 10 | 0 | 0 | 0 |
| 15 | 4 | 10 | 0 | 0 | 0 |
| 0 | 0 | 0 | 7 | 0 | 10 |
| 0 | 0 | 0 | 15 | 8 | 16 |
| 0 | 0 | 0 | 13 | 11 | 4 |
G:=sub<GL(6,GF(19))| [11,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,11],[7,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[7,18,7,0,0,0,0,1,0,0,0,0,0,0,11,0,0,0,0,0,0,7,0,11,0,0,0,0,7,16,0,0,0,0,0,1],[1,0,0,0,0,0,6,18,18,0,0,0,0,1,0,0,0,0,0,0,0,1,0,8,0,0,0,11,11,15,0,0,0,0,0,7],[9,9,15,0,0,0,4,0,4,0,0,0,6,10,10,0,0,0,0,0,0,7,15,13,0,0,0,0,8,11,0,0,0,10,16,4] >;
C32.24He3 in GAP, Magma, Sage, TeX
C_3^2._{24}{\rm He}_3 % in TeX
G:=Group("C3^2.24He3"); // GroupNames label
G:=SmallGroup(243,3);
// by ID
G=gap.SmallGroup(243,3);
# by ID
G:=PCGroup([5,-3,3,-3,-3,3,121,542,457]);
// 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,d*c*d^-1=b*c=c*b,b*d=d*b,b*e=e*b,e*c*e^-1=a*c*d^-1,e*d*e^-1=a^-1*d>;
// generators/relations
Export