p-group, metabelian, nilpotent (class 3), monomial
Aliases: C32.28He3, C33.5C32, C3.8C3≀C3, C32⋊C9.2C3, C3.4(C3.He3), (C3×3- 1+2).2C3, SmallGroup(243,7)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C32.28He3
G = < a,b,c,d,e | a3=b3=d3=1, c3=a-1, e3=a, ab=ba, ac=ca, ad=da, ae=ea, dcd-1=bc=cb, bd=db, be=eb, ece-1=acd-1, ede-1=a-1d >
Smallest permutation representation of C32.28He3
►On 81 points
Generators in S81
(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 40 30)(2 41 31)(3 42 32)(4 43 33)(5 44 34)(6 45 35)(7 37 36)(8 38 28)(9 39 29)(10 78 23)(11 79 24)(12 80 25)(13 81 26)(14 73 27)(15 74 19)(16 75 20)(17 76 21)(18 77 22)(46 62 66)(47 63 67)(48 55 68)(49 56 69)(50 57 70)(51 58 71)(52 59 72)(53 60 64)(54 61 65)
(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)
(2 31 41)(3 42 32)(5 34 44)(6 45 35)(8 28 38)(9 39 29)(10 81 20)(11 14 17)(12 19 77)(13 75 23)(15 22 80)(16 78 26)(18 25 74)(21 24 27)(46 52 49)(47 64 57)(48 61 71)(50 67 60)(51 55 65)(53 70 63)(54 58 68)(56 62 59)(66 72 69)(73 76 79)
(1 21 49 7 27 46 4 24 52)(2 80 57 8 77 63 5 74 60)(3 20 51 9 26 48 6 23 54)(10 61 42 16 58 39 13 55 45)(11 59 40 17 56 37 14 62 43)(12 50 28 18 47 34 15 53 31)(19 64 41 25 70 38 22 67 44)(29 81 68 35 78 65 32 75 71)(30 76 69 36 73 66 33 79 72)
G:=sub<Sym(81)| (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), (1,40,30)(2,41,31)(3,42,32)(4,43,33)(5,44,34)(6,45,35)(7,37,36)(8,38,28)(9,39,29)(10,78,23)(11,79,24)(12,80,25)(13,81,26)(14,73,27)(15,74,19)(16,75,20)(17,76,21)(18,77,22)(46,62,66)(47,63,67)(48,55,68)(49,56,69)(50,57,70)(51,58,71)(52,59,72)(53,60,64)(54,61,65), (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), (2,31,41)(3,42,32)(5,34,44)(6,45,35)(8,28,38)(9,39,29)(10,81,20)(11,14,17)(12,19,77)(13,75,23)(15,22,80)(16,78,26)(18,25,74)(21,24,27)(46,52,49)(47,64,57)(48,61,71)(50,67,60)(51,55,65)(53,70,63)(54,58,68)(56,62,59)(66,72,69)(73,76,79), (1,21,49,7,27,46,4,24,52)(2,80,57,8,77,63,5,74,60)(3,20,51,9,26,48,6,23,54)(10,61,42,16,58,39,13,55,45)(11,59,40,17,56,37,14,62,43)(12,50,28,18,47,34,15,53,31)(19,64,41,25,70,38,22,67,44)(29,81,68,35,78,65,32,75,71)(30,76,69,36,73,66,33,79,72)>;
G:=Group( (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), (1,40,30)(2,41,31)(3,42,32)(4,43,33)(5,44,34)(6,45,35)(7,37,36)(8,38,28)(9,39,29)(10,78,23)(11,79,24)(12,80,25)(13,81,26)(14,73,27)(15,74,19)(16,75,20)(17,76,21)(18,77,22)(46,62,66)(47,63,67)(48,55,68)(49,56,69)(50,57,70)(51,58,71)(52,59,72)(53,60,64)(54,61,65), (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), (2,31,41)(3,42,32)(5,34,44)(6,45,35)(8,28,38)(9,39,29)(10,81,20)(11,14,17)(12,19,77)(13,75,23)(15,22,80)(16,78,26)(18,25,74)(21,24,27)(46,52,49)(47,64,57)(48,61,71)(50,67,60)(51,55,65)(53,70,63)(54,58,68)(56,62,59)(66,72,69)(73,76,79), (1,21,49,7,27,46,4,24,52)(2,80,57,8,77,63,5,74,60)(3,20,51,9,26,48,6,23,54)(10,61,42,16,58,39,13,55,45)(11,59,40,17,56,37,14,62,43)(12,50,28,18,47,34,15,53,31)(19,64,41,25,70,38,22,67,44)(29,81,68,35,78,65,32,75,71)(30,76,69,36,73,66,33,79,72) );
G=PermutationGroup([[(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)], [(1,40,30),(2,41,31),(3,42,32),(4,43,33),(5,44,34),(6,45,35),(7,37,36),(8,38,28),(9,39,29),(10,78,23),(11,79,24),(12,80,25),(13,81,26),(14,73,27),(15,74,19),(16,75,20),(17,76,21),(18,77,22),(46,62,66),(47,63,67),(48,55,68),(49,56,69),(50,57,70),(51,58,71),(52,59,72),(53,60,64),(54,61,65)], [(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)], [(2,31,41),(3,42,32),(5,34,44),(6,45,35),(8,28,38),(9,39,29),(10,81,20),(11,14,17),(12,19,77),(13,75,23),(15,22,80),(16,78,26),(18,25,74),(21,24,27),(46,52,49),(47,64,57),(48,61,71),(50,67,60),(51,55,65),(53,70,63),(54,58,68),(56,62,59),(66,72,69),(73,76,79)], [(1,21,49,7,27,46,4,24,52),(2,80,57,8,77,63,5,74,60),(3,20,51,9,26,48,6,23,54),(10,61,42,16,58,39,13,55,45),(11,59,40,17,56,37,14,62,43),(12,50,28,18,47,34,15,53,31),(19,64,41,25,70,38,22,67,44),(29,81,68,35,78,65,32,75,71),(30,76,69,36,73,66,33,79,72)]])
C32.28He3 is a maximal subgroup of
C3.(C33⋊S3)
35 conjugacy classes
| class | 1 | 3A | ··· | 3H | 3I | 3J | 9A | ··· | 9X |
| 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 | C3.He3 |
| kernel | C32.28He3 | C32⋊C9 | C3×3- 1+2 | C32 | C3 | C3 |
| # reps | 1 | 4 | 4 | 2 | 12 | 12 |
Matrix representation of C32.28He3 ►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 | 11 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 0 |
| 0 | 0 | 0 | 0 | 0 | 11 |
| 7 | 6 | 0 | 0 | 0 | 0 |
| 0 | 12 | 1 | 0 | 0 | 0 |
| 1 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 7 | 10 | 0 |
| 0 | 0 | 0 | 0 | 12 | 1 |
| 0 | 0 | 0 | 12 | 8 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 7 | 7 | 0 | 0 | 0 | 0 |
| 12 | 0 | 11 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 7 | 11 | 0 |
| 0 | 0 | 0 | 18 | 0 | 7 |
| 4 | 0 | 16 | 0 | 0 | 0 |
| 9 | 0 | 13 | 0 | 0 | 0 |
| 0 | 9 | 15 | 0 | 0 | 0 |
| 0 | 0 | 0 | 7 | 10 | 0 |
| 0 | 0 | 0 | 0 | 12 | 1 |
| 0 | 0 | 0 | 1 | 8 | 0 |
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],[7,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,11],[7,0,1,0,0,0,6,12,8,0,0,0,0,1,0,0,0,0,0,0,0,7,0,12,0,0,0,10,12,8,0,0,0,0,1,0],[1,7,12,0,0,0,0,7,0,0,0,0,0,0,11,0,0,0,0,0,0,1,7,18,0,0,0,0,11,0,0,0,0,0,0,7],[4,9,0,0,0,0,0,0,9,0,0,0,16,13,15,0,0,0,0,0,0,7,0,1,0,0,0,10,12,8,0,0,0,0,1,0] >;
C32.28He3 in GAP, Magma, Sage, TeX
C_3^2._{28}{\rm He}_3 % in TeX
G:=Group("C3^2.28He3"); // GroupNames label
G:=SmallGroup(243,7);
// by ID
G=gap.SmallGroup(243,7);
# by ID
G:=PCGroup([5,-3,3,-3,-3,3,135,121,186,542,457]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^3=d^3=1,c^3=a^-1,e^3=a,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