p-group, metabelian, nilpotent (class 3), monomial
Aliases: C32.20He3, C33.20C32, C32.63- 1+2, (C3×C9)⋊3C9, C32⋊C9.6C3, C32.9(C3×C9), (C32×C9).4C3, C3.5(C32⋊C9), C3.1(He3⋊C3), C3.1(C3.He3), SmallGroup(243,15)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C32.20He3
G = < a,b,c,d,e | a3=b3=d3=1, c3=b-1, e3=a, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=bcd-1, ede-1=b-1d >
Smallest permutation representation of C32.20He3
►On 81 points
Generators in S81
(1 38 35)(2 39 36)(3 40 28)(4 41 29)(5 42 30)(6 43 31)(7 44 32)(8 45 33)(9 37 34)(10 76 23)(11 77 24)(12 78 25)(13 79 26)(14 80 27)(15 81 19)(16 73 20)(17 74 21)(18 75 22)(46 60 68)(47 61 69)(48 62 70)(49 63 71)(50 55 72)(51 56 64)(52 57 65)(53 58 66)(54 59 67)
(1 7 4)(2 8 5)(3 9 6)(10 16 13)(11 17 14)(12 18 15)(19 25 22)(20 26 23)(21 27 24)(28 34 31)(29 35 32)(30 36 33)(37 43 40)(38 44 41)(39 45 42)(46 52 49)(47 53 50)(48 54 51)(55 61 58)(56 62 59)(57 63 60)(64 70 67)(65 71 68)(66 72 69)(73 79 76)(74 80 77)(75 81 78)
(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 38 35)(2 39 36)(3 40 28)(4 41 29)(5 42 30)(6 43 31)(7 44 32)(8 45 33)(9 37 34)(10 79 20)(11 80 21)(12 81 22)(13 73 23)(14 74 24)(15 75 25)(16 76 26)(17 77 27)(18 78 19)(46 57 71)(47 58 72)(48 59 64)(49 60 65)(50 61 66)(51 62 67)(52 63 68)(53 55 69)(54 56 70)
(1 75 48 38 22 62 35 18 70)(2 26 71 39 13 49 36 79 63)(3 17 55 40 74 72 28 21 50)(4 78 51 41 25 56 29 12 64)(5 20 65 42 16 52 30 73 57)(6 11 58 43 77 66 31 24 53)(7 81 54 44 19 59 32 15 67)(8 23 68 45 10 46 33 76 60)(9 14 61 37 80 69 34 27 47)
G:=sub<Sym(81)| (1,38,35)(2,39,36)(3,40,28)(4,41,29)(5,42,30)(6,43,31)(7,44,32)(8,45,33)(9,37,34)(10,76,23)(11,77,24)(12,78,25)(13,79,26)(14,80,27)(15,81,19)(16,73,20)(17,74,21)(18,75,22)(46,60,68)(47,61,69)(48,62,70)(49,63,71)(50,55,72)(51,56,64)(52,57,65)(53,58,66)(54,59,67), (1,7,4)(2,8,5)(3,9,6)(10,16,13)(11,17,14)(12,18,15)(19,25,22)(20,26,23)(21,27,24)(28,34,31)(29,35,32)(30,36,33)(37,43,40)(38,44,41)(39,45,42)(46,52,49)(47,53,50)(48,54,51)(55,61,58)(56,62,59)(57,63,60)(64,70,67)(65,71,68)(66,72,69)(73,79,76)(74,80,77)(75,81,78), (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), (1,38,35)(2,39,36)(3,40,28)(4,41,29)(5,42,30)(6,43,31)(7,44,32)(8,45,33)(9,37,34)(10,79,20)(11,80,21)(12,81,22)(13,73,23)(14,74,24)(15,75,25)(16,76,26)(17,77,27)(18,78,19)(46,57,71)(47,58,72)(48,59,64)(49,60,65)(50,61,66)(51,62,67)(52,63,68)(53,55,69)(54,56,70), (1,75,48,38,22,62,35,18,70)(2,26,71,39,13,49,36,79,63)(3,17,55,40,74,72,28,21,50)(4,78,51,41,25,56,29,12,64)(5,20,65,42,16,52,30,73,57)(6,11,58,43,77,66,31,24,53)(7,81,54,44,19,59,32,15,67)(8,23,68,45,10,46,33,76,60)(9,14,61,37,80,69,34,27,47)>;
G:=Group( (1,38,35)(2,39,36)(3,40,28)(4,41,29)(5,42,30)(6,43,31)(7,44,32)(8,45,33)(9,37,34)(10,76,23)(11,77,24)(12,78,25)(13,79,26)(14,80,27)(15,81,19)(16,73,20)(17,74,21)(18,75,22)(46,60,68)(47,61,69)(48,62,70)(49,63,71)(50,55,72)(51,56,64)(52,57,65)(53,58,66)(54,59,67), (1,7,4)(2,8,5)(3,9,6)(10,16,13)(11,17,14)(12,18,15)(19,25,22)(20,26,23)(21,27,24)(28,34,31)(29,35,32)(30,36,33)(37,43,40)(38,44,41)(39,45,42)(46,52,49)(47,53,50)(48,54,51)(55,61,58)(56,62,59)(57,63,60)(64,70,67)(65,71,68)(66,72,69)(73,79,76)(74,80,77)(75,81,78), (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), (1,38,35)(2,39,36)(3,40,28)(4,41,29)(5,42,30)(6,43,31)(7,44,32)(8,45,33)(9,37,34)(10,79,20)(11,80,21)(12,81,22)(13,73,23)(14,74,24)(15,75,25)(16,76,26)(17,77,27)(18,78,19)(46,57,71)(47,58,72)(48,59,64)(49,60,65)(50,61,66)(51,62,67)(52,63,68)(53,55,69)(54,56,70), (1,75,48,38,22,62,35,18,70)(2,26,71,39,13,49,36,79,63)(3,17,55,40,74,72,28,21,50)(4,78,51,41,25,56,29,12,64)(5,20,65,42,16,52,30,73,57)(6,11,58,43,77,66,31,24,53)(7,81,54,44,19,59,32,15,67)(8,23,68,45,10,46,33,76,60)(9,14,61,37,80,69,34,27,47) );
G=PermutationGroup([[(1,38,35),(2,39,36),(3,40,28),(4,41,29),(5,42,30),(6,43,31),(7,44,32),(8,45,33),(9,37,34),(10,76,23),(11,77,24),(12,78,25),(13,79,26),(14,80,27),(15,81,19),(16,73,20),(17,74,21),(18,75,22),(46,60,68),(47,61,69),(48,62,70),(49,63,71),(50,55,72),(51,56,64),(52,57,65),(53,58,66),(54,59,67)], [(1,7,4),(2,8,5),(3,9,6),(10,16,13),(11,17,14),(12,18,15),(19,25,22),(20,26,23),(21,27,24),(28,34,31),(29,35,32),(30,36,33),(37,43,40),(38,44,41),(39,45,42),(46,52,49),(47,53,50),(48,54,51),(55,61,58),(56,62,59),(57,63,60),(64,70,67),(65,71,68),(66,72,69),(73,79,76),(74,80,77),(75,81,78)], [(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)], [(1,38,35),(2,39,36),(3,40,28),(4,41,29),(5,42,30),(6,43,31),(7,44,32),(8,45,33),(9,37,34),(10,79,20),(11,80,21),(12,81,22),(13,73,23),(14,74,24),(15,75,25),(16,76,26),(17,77,27),(18,78,19),(46,57,71),(47,58,72),(48,59,64),(49,60,65),(50,61,66),(51,62,67),(52,63,68),(53,55,69),(54,56,70)], [(1,75,48,38,22,62,35,18,70),(2,26,71,39,13,49,36,79,63),(3,17,55,40,74,72,28,21,50),(4,78,51,41,25,56,29,12,64),(5,20,65,42,16,52,30,73,57),(6,11,58,43,77,66,31,24,53),(7,81,54,44,19,59,32,15,67),(8,23,68,45,10,46,33,76,60),(9,14,61,37,80,69,34,27,47)]])
C32.20He3 is a maximal subgroup of
C9⋊S3⋊3C9 (C3×C9)⋊3D9 (C3×C9)⋊6D9
51 conjugacy classes
| class | 1 | 3A | ··· | 3H | 3I | ··· | 3N | 9A | ··· | 9R | 9S | ··· | 9AJ |
| order | 1 | 3 | ··· | 3 | 3 | ··· | 3 | 9 | ··· | 9 | 9 | ··· | 9 |
| size | 1 | 1 | ··· | 1 | 3 | ··· | 3 | 3 | ··· | 3 | 9 | ··· | 9 |
51 irreducible representations
| dim | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 |
| type | + | |||||||
| image | C1 | C3 | C3 | C9 | He3 | 3- 1+2 | He3⋊C3 | C3.He3 |
| kernel | C32.20He3 | C32⋊C9 | C32×C9 | C3×C9 | C32 | C32 | C3 | C3 |
| # reps | 1 | 6 | 2 | 18 | 2 | 4 | 6 | 12 |
Matrix representation of C32.20He3 ►in GL4(𝔽19) generated by
| 11 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 11 | 0 | 0 |
| 0 | 0 | 11 | 0 |
| 0 | 0 | 0 | 11 |
| 11 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 4 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 11 | 0 |
| 0 | 0 | 0 | 7 |
| 5 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
G:=sub<GL(4,GF(19))| [11,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,11,0,0,0,0,11,0,0,0,0,11],[11,0,0,0,0,9,0,0,0,0,9,0,0,0,0,4],[1,0,0,0,0,1,0,0,0,0,11,0,0,0,0,7],[5,0,0,0,0,0,0,1,0,1,0,0,0,0,1,0] >;
C32.20He3 in GAP, Magma, Sage, TeX
C_3^2._{20}{\rm He}_3 % in TeX
G:=Group("C3^2.20He3"); // GroupNames label
G:=SmallGroup(243,15);
// by ID
G=gap.SmallGroup(243,15);
# by ID
G:=PCGroup([5,-3,3,-3,3,-3,135,121,546,1352]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^3=d^3=1,c^3=b^-1,e^3=a,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=b*c*d^-1,e*d*e^-1=b^-1*d>;
// generators/relations
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