p-group, metabelian, nilpotent (class 3), monomial
Aliases: C33.3C32, C32.26He3, C3.6C3≀C3, C32⋊C9.1C3, C3.5(He3.C3), C3.3(C3.He3), (C3×3- 1+2).1C3, SmallGroup(243,5)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C33.3C32
G = < a,b,c,d,e | a3=b3=d3=1, c3=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 C33.3C32
►On 81 points
Generators in S81
(1 4 7)(2 5 8)(3 6 9)(10 13 16)(11 14 17)(12 15 18)(19 22 25)(20 23 26)(21 24 27)(28 31 34)(29 32 35)(30 33 36)(37 40 43)(38 41 44)(39 42 45)(46 49 52)(47 50 53)(48 51 54)(55 58 61)(56 59 62)(57 60 63)(64 67 70)(65 68 71)(66 69 72)(73 76 79)(74 77 80)(75 78 81)
(1 43 36)(2 44 28)(3 45 29)(4 37 30)(5 38 31)(6 39 32)(7 40 33)(8 41 34)(9 42 35)(10 77 23)(11 78 24)(12 79 25)(13 80 26)(14 81 27)(15 73 19)(16 74 20)(17 75 21)(18 76 22)(46 62 72)(47 63 64)(48 55 65)(49 56 66)(50 57 67)(51 58 68)(52 59 69)(53 60 70)(54 61 71)
(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 28 44)(3 45 29)(5 31 38)(6 39 32)(8 34 41)(9 42 35)(10 74 26)(11 17 14)(12 22 73)(13 77 20)(15 25 76)(16 80 23)(18 19 79)(21 27 24)(46 49 52)(47 67 60)(48 58 71)(50 70 63)(51 61 65)(53 64 57)(54 55 68)(56 59 62)(66 69 72)(75 81 78)
(1 21 52 4 24 46 7 27 49)(2 73 60 5 76 63 8 79 57)(3 26 54 6 20 48 9 23 51)(10 58 45 13 61 39 16 55 42)(11 62 40 14 56 43 17 59 37)(12 50 28 15 53 31 18 47 34)(19 70 38 22 64 41 25 67 44)(29 80 71 32 74 65 35 77 68)(30 78 72 33 81 66 36 75 69)
G:=sub<Sym(81)| (1,4,7)(2,5,8)(3,6,9)(10,13,16)(11,14,17)(12,15,18)(19,22,25)(20,23,26)(21,24,27)(28,31,34)(29,32,35)(30,33,36)(37,40,43)(38,41,44)(39,42,45)(46,49,52)(47,50,53)(48,51,54)(55,58,61)(56,59,62)(57,60,63)(64,67,70)(65,68,71)(66,69,72)(73,76,79)(74,77,80)(75,78,81), (1,43,36)(2,44,28)(3,45,29)(4,37,30)(5,38,31)(6,39,32)(7,40,33)(8,41,34)(9,42,35)(10,77,23)(11,78,24)(12,79,25)(13,80,26)(14,81,27)(15,73,19)(16,74,20)(17,75,21)(18,76,22)(46,62,72)(47,63,64)(48,55,65)(49,56,66)(50,57,67)(51,58,68)(52,59,69)(53,60,70)(54,61,71), (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), (2,28,44)(3,45,29)(5,31,38)(6,39,32)(8,34,41)(9,42,35)(10,74,26)(11,17,14)(12,22,73)(13,77,20)(15,25,76)(16,80,23)(18,19,79)(21,27,24)(46,49,52)(47,67,60)(48,58,71)(50,70,63)(51,61,65)(53,64,57)(54,55,68)(56,59,62)(66,69,72)(75,81,78), (1,21,52,4,24,46,7,27,49)(2,73,60,5,76,63,8,79,57)(3,26,54,6,20,48,9,23,51)(10,58,45,13,61,39,16,55,42)(11,62,40,14,56,43,17,59,37)(12,50,28,15,53,31,18,47,34)(19,70,38,22,64,41,25,67,44)(29,80,71,32,74,65,35,77,68)(30,78,72,33,81,66,36,75,69)>;
G:=Group( (1,4,7)(2,5,8)(3,6,9)(10,13,16)(11,14,17)(12,15,18)(19,22,25)(20,23,26)(21,24,27)(28,31,34)(29,32,35)(30,33,36)(37,40,43)(38,41,44)(39,42,45)(46,49,52)(47,50,53)(48,51,54)(55,58,61)(56,59,62)(57,60,63)(64,67,70)(65,68,71)(66,69,72)(73,76,79)(74,77,80)(75,78,81), (1,43,36)(2,44,28)(3,45,29)(4,37,30)(5,38,31)(6,39,32)(7,40,33)(8,41,34)(9,42,35)(10,77,23)(11,78,24)(12,79,25)(13,80,26)(14,81,27)(15,73,19)(16,74,20)(17,75,21)(18,76,22)(46,62,72)(47,63,64)(48,55,65)(49,56,66)(50,57,67)(51,58,68)(52,59,69)(53,60,70)(54,61,71), (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), (2,28,44)(3,45,29)(5,31,38)(6,39,32)(8,34,41)(9,42,35)(10,74,26)(11,17,14)(12,22,73)(13,77,20)(15,25,76)(16,80,23)(18,19,79)(21,27,24)(46,49,52)(47,67,60)(48,58,71)(50,70,63)(51,61,65)(53,64,57)(54,55,68)(56,59,62)(66,69,72)(75,81,78), (1,21,52,4,24,46,7,27,49)(2,73,60,5,76,63,8,79,57)(3,26,54,6,20,48,9,23,51)(10,58,45,13,61,39,16,55,42)(11,62,40,14,56,43,17,59,37)(12,50,28,15,53,31,18,47,34)(19,70,38,22,64,41,25,67,44)(29,80,71,32,74,65,35,77,68)(30,78,72,33,81,66,36,75,69) );
G=PermutationGroup([[(1,4,7),(2,5,8),(3,6,9),(10,13,16),(11,14,17),(12,15,18),(19,22,25),(20,23,26),(21,24,27),(28,31,34),(29,32,35),(30,33,36),(37,40,43),(38,41,44),(39,42,45),(46,49,52),(47,50,53),(48,51,54),(55,58,61),(56,59,62),(57,60,63),(64,67,70),(65,68,71),(66,69,72),(73,76,79),(74,77,80),(75,78,81)], [(1,43,36),(2,44,28),(3,45,29),(4,37,30),(5,38,31),(6,39,32),(7,40,33),(8,41,34),(9,42,35),(10,77,23),(11,78,24),(12,79,25),(13,80,26),(14,81,27),(15,73,19),(16,74,20),(17,75,21),(18,76,22),(46,62,72),(47,63,64),(48,55,65),(49,56,66),(50,57,67),(51,58,68),(52,59,69),(53,60,70),(54,61,71)], [(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)], [(2,28,44),(3,45,29),(5,31,38),(6,39,32),(8,34,41),(9,42,35),(10,74,26),(11,17,14),(12,22,73),(13,77,20),(15,25,76),(16,80,23),(18,19,79),(21,27,24),(46,49,52),(47,67,60),(48,58,71),(50,70,63),(51,61,65),(53,64,57),(54,55,68),(56,59,62),(66,69,72),(75,81,78)], [(1,21,52,4,24,46,7,27,49),(2,73,60,5,76,63,8,79,57),(3,26,54,6,20,48,9,23,51),(10,58,45,13,61,39,16,55,42),(11,62,40,14,56,43,17,59,37),(12,50,28,15,53,31,18,47,34),(19,70,38,22,64,41,25,67,44),(29,80,71,32,74,65,35,77,68),(30,78,72,33,81,66,36,75,69)]])
C33.3C32 is a maximal subgroup of
C33.(C3⋊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 | 3 |
| type | + | ||||||
| image | C1 | C3 | C3 | He3 | C3≀C3 | He3.C3 | C3.He3 |
| kernel | C33.3C32 | C32⋊C9 | C3×3- 1+2 | C32 | C3 | C3 | C3 |
| # reps | 1 | 6 | 2 | 2 | 6 | 12 | 6 |
Matrix representation of C33.3C32 ►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 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 11 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 17 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 0 |
| 0 | 0 | 0 | 17 | 14 | 16 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 10 | 11 | 0 |
| 0 | 0 | 0 | 0 | 18 | 7 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 5 | 0 | 0 | 0 | 0 | 0 |
| 0 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 11 | 11 | 0 |
| 0 | 0 | 0 | 15 | 16 | 13 |
| 0 | 0 | 0 | 1 | 12 | 11 |
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],[0,0,11,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,17,0,17,0,0,0,0,17,14,0,0,0,0,0,16],[1,0,0,0,0,0,0,7,0,0,0,0,0,0,11,0,0,0,0,0,0,1,10,0,0,0,0,0,11,18,0,0,0,0,0,7],[0,5,0,0,0,0,0,0,17,0,0,0,16,0,0,0,0,0,0,0,0,11,15,1,0,0,0,11,16,12,0,0,0,0,13,11] >;
C33.3C32 in GAP, Magma, Sage, TeX
C_3^3._3C_3^2
% in TeX
G:=Group("C3^3.3C3^2"); // GroupNames label
G:=SmallGroup(243,5);
// by ID
G=gap.SmallGroup(243,5);
# by ID
G:=PCGroup([5,-3,3,-3,-3,3,135,121,96,542,457]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^3=d^3=1,c^3=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
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