p-group, metabelian, nilpotent (class 3), monomial
Aliases: C33.7C32, C32.30He3, C32⋊C9.4C3, C3.8(He3.C3), 3-Sylow(J3), SmallGroup(243,9)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C33.7C32
G = < a,b,c,d,e | a3=b3=c3=1, d3=c-1, e3=b, ab=ba, ac=ca, dad-1=ab-1, eae-1=ac-1, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=ab-1c-1d >
Smallest permutation representation of C33.7C32
►On 81 points
Generators in S81
(2 48 78)(3 79 49)(5 51 81)(6 73 52)(8 54 75)(9 76 46)(10 13 16)(11 27 56)(12 63 22)(14 21 59)(15 57 25)(17 24 62)(18 60 19)(20 23 26)(28 43 67)(29 35 32)(30 72 42)(31 37 70)(33 66 45)(34 40 64)(36 69 39)(38 44 41)(55 58 61)(65 71 68)
(1 77 47)(2 78 48)(3 79 49)(4 80 50)(5 81 51)(6 73 52)(7 74 53)(8 75 54)(9 76 46)(10 58 23)(11 59 24)(12 60 25)(13 61 26)(14 62 27)(15 63 19)(16 55 20)(17 56 21)(18 57 22)(28 37 64)(29 38 65)(30 39 66)(31 40 67)(32 41 68)(33 42 69)(34 43 70)(35 44 71)(36 45 72)
(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 61 35 77 26 44 47 13 71)(2 24 72 78 11 36 48 59 45)(3 57 28 79 22 37 49 18 64)(4 55 29 80 20 38 50 16 65)(5 27 66 81 14 30 51 62 39)(6 60 31 73 25 40 52 12 67)(7 58 32 74 23 41 53 10 68)(8 21 69 75 17 33 54 56 42)(9 63 34 76 19 43 46 15 70)
G:=sub<Sym(81)| (2,48,78)(3,79,49)(5,51,81)(6,73,52)(8,54,75)(9,76,46)(10,13,16)(11,27,56)(12,63,22)(14,21,59)(15,57,25)(17,24,62)(18,60,19)(20,23,26)(28,43,67)(29,35,32)(30,72,42)(31,37,70)(33,66,45)(34,40,64)(36,69,39)(38,44,41)(55,58,61)(65,71,68), (1,77,47)(2,78,48)(3,79,49)(4,80,50)(5,81,51)(6,73,52)(7,74,53)(8,75,54)(9,76,46)(10,58,23)(11,59,24)(12,60,25)(13,61,26)(14,62,27)(15,63,19)(16,55,20)(17,56,21)(18,57,22)(28,37,64)(29,38,65)(30,39,66)(31,40,67)(32,41,68)(33,42,69)(34,43,70)(35,44,71)(36,45,72), (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), (1,61,35,77,26,44,47,13,71)(2,24,72,78,11,36,48,59,45)(3,57,28,79,22,37,49,18,64)(4,55,29,80,20,38,50,16,65)(5,27,66,81,14,30,51,62,39)(6,60,31,73,25,40,52,12,67)(7,58,32,74,23,41,53,10,68)(8,21,69,75,17,33,54,56,42)(9,63,34,76,19,43,46,15,70)>;
G:=Group( (2,48,78)(3,79,49)(5,51,81)(6,73,52)(8,54,75)(9,76,46)(10,13,16)(11,27,56)(12,63,22)(14,21,59)(15,57,25)(17,24,62)(18,60,19)(20,23,26)(28,43,67)(29,35,32)(30,72,42)(31,37,70)(33,66,45)(34,40,64)(36,69,39)(38,44,41)(55,58,61)(65,71,68), (1,77,47)(2,78,48)(3,79,49)(4,80,50)(5,81,51)(6,73,52)(7,74,53)(8,75,54)(9,76,46)(10,58,23)(11,59,24)(12,60,25)(13,61,26)(14,62,27)(15,63,19)(16,55,20)(17,56,21)(18,57,22)(28,37,64)(29,38,65)(30,39,66)(31,40,67)(32,41,68)(33,42,69)(34,43,70)(35,44,71)(36,45,72), (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), (1,61,35,77,26,44,47,13,71)(2,24,72,78,11,36,48,59,45)(3,57,28,79,22,37,49,18,64)(4,55,29,80,20,38,50,16,65)(5,27,66,81,14,30,51,62,39)(6,60,31,73,25,40,52,12,67)(7,58,32,74,23,41,53,10,68)(8,21,69,75,17,33,54,56,42)(9,63,34,76,19,43,46,15,70) );
G=PermutationGroup([[(2,48,78),(3,79,49),(5,51,81),(6,73,52),(8,54,75),(9,76,46),(10,13,16),(11,27,56),(12,63,22),(14,21,59),(15,57,25),(17,24,62),(18,60,19),(20,23,26),(28,43,67),(29,35,32),(30,72,42),(31,37,70),(33,66,45),(34,40,64),(36,69,39),(38,44,41),(55,58,61),(65,71,68)], [(1,77,47),(2,78,48),(3,79,49),(4,80,50),(5,81,51),(6,73,52),(7,74,53),(8,75,54),(9,76,46),(10,58,23),(11,59,24),(12,60,25),(13,61,26),(14,62,27),(15,63,19),(16,55,20),(17,56,21),(18,57,22),(28,37,64),(29,38,65),(30,39,66),(31,40,67),(32,41,68),(33,42,69),(34,43,70),(35,44,71),(36,45,72)], [(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)], [(1,61,35,77,26,44,47,13,71),(2,24,72,78,11,36,48,59,45),(3,57,28,79,22,37,49,18,64),(4,55,29,80,20,38,50,16,65),(5,27,66,81,14,30,51,62,39),(6,60,31,73,25,40,52,12,67),(7,58,32,74,23,41,53,10,68),(8,21,69,75,17,33,54,56,42),(9,63,34,76,19,43,46,15,70)]])
C33.7C32 is a maximal subgroup of
C32⋊2D9.C3 C32⋊C9.10S3
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 | 3 | 3 |
| type | + | |||
| image | C1 | C3 | He3 | He3.C3 |
| kernel | C33.7C32 | C32⋊C9 | C32 | C3 |
| # reps | 1 | 8 | 2 | 24 |
Matrix representation of C33.7C32 ►in GL6(𝔽19)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 5 | 7 | 0 |
| 0 | 0 | 0 | 10 | 0 | 11 |
| 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 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 0 |
| 0 | 0 | 0 | 0 | 0 | 7 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 11 | 0 | 11 |
| 0 | 0 | 0 | 0 | 0 | 13 |
| 0 | 0 | 0 | 15 | 16 | 8 |
| 5 | 0 | 0 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 17 | 0 | 0 | 0 |
| 0 | 0 | 0 | 5 | 6 | 0 |
| 0 | 0 | 0 | 0 | 14 | 1 |
| 0 | 0 | 0 | 0 | 13 | 0 |
G:=sub<GL(6,GF(19))| [1,0,0,0,0,0,0,11,0,0,0,0,0,0,7,0,0,0,0,0,0,1,5,10,0,0,0,0,7,0,0,0,0,0,0,11],[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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,7],[0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,11,0,15,0,0,0,0,0,16,0,0,0,11,13,8],[5,0,0,0,0,0,0,5,0,0,0,0,0,0,17,0,0,0,0,0,0,5,0,0,0,0,0,6,14,13,0,0,0,0,1,0] >;
C33.7C32 in GAP, Magma, Sage, TeX
C_3^3._7C_3^2
% in TeX
G:=Group("C3^3.7C3^2"); // GroupNames label
G:=SmallGroup(243,9);
// by ID
G=gap.SmallGroup(243,9);
# by ID
G:=PCGroup([5,-3,3,-3,-3,3,810,121,96,542,457]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^3=c^3=1,d^3=c^-1,e^3=b,a*b=b*a,a*c=c*a,d*a*d^-1=a*b^-1,e*a*e^-1=a*c^-1,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=a*b^-1*c^-1*d>;
// generators/relations
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