p-group, metabelian, nilpotent (class 3), monomial
Aliases: C33.2C32, C32.25He3, C3.5C3≀C3, C32⋊C9⋊2C3, (C3×He3).1C3, C3.4(He3.C3), (C3×3- 1+2)⋊1C3, SmallGroup(243,4)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C33.C32
G = < a,b,c,d,e | a3=b3=c3=d3=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 >
9C3
9C3
9C3
9C3
3C9
3C9
3C32
3C32
3C9
3C9
3C9
3C32
3C32
3C9
3C32
3C32
3C32
9C9
9C32
9C32
9C32
3He3
3He3
3He3
3C33
Smallest permutation representation of C33.C32
►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 80 26)(2 81 27)(3 73 19)(4 74 20)(5 75 21)(6 76 22)(7 77 23)(8 78 24)(9 79 25)(10 34 54)(11 35 46)(12 36 47)(13 28 48)(14 29 49)(15 30 50)(16 31 51)(17 32 52)(18 33 53)(37 61 71)(38 62 72)(39 63 64)(40 55 65)(41 56 66)(42 57 67)(43 58 68)(44 59 69)(45 60 70)
(1 7 4)(2 10 71)(3 44 29)(5 13 65)(6 38 32)(8 16 68)(9 41 35)(11 25 66)(12 53 30)(14 19 69)(15 47 33)(17 22 72)(18 50 36)(20 26 23)(21 48 55)(24 51 58)(27 54 61)(28 40 75)(31 43 78)(34 37 81)(39 60 67)(42 63 70)(45 57 64)(46 79 56)(49 73 59)(52 76 62)(74 80 77)
(1 45 18)(2 43 13)(3 41 17)(4 39 12)(5 37 16)(6 44 11)(7 42 15)(8 40 10)(9 38 14)(19 66 52)(20 64 47)(21 71 51)(22 69 46)(23 67 50)(24 65 54)(25 72 49)(26 70 53)(27 68 48)(28 81 58)(29 79 62)(30 77 57)(31 75 61)(32 73 56)(33 80 60)(34 78 55)(35 76 59)(36 74 63)
(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)
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), (1,80,26)(2,81,27)(3,73,19)(4,74,20)(5,75,21)(6,76,22)(7,77,23)(8,78,24)(9,79,25)(10,34,54)(11,35,46)(12,36,47)(13,28,48)(14,29,49)(15,30,50)(16,31,51)(17,32,52)(18,33,53)(37,61,71)(38,62,72)(39,63,64)(40,55,65)(41,56,66)(42,57,67)(43,58,68)(44,59,69)(45,60,70), (1,7,4)(2,10,71)(3,44,29)(5,13,65)(6,38,32)(8,16,68)(9,41,35)(11,25,66)(12,53,30)(14,19,69)(15,47,33)(17,22,72)(18,50,36)(20,26,23)(21,48,55)(24,51,58)(27,54,61)(28,40,75)(31,43,78)(34,37,81)(39,60,67)(42,63,70)(45,57,64)(46,79,56)(49,73,59)(52,76,62)(74,80,77), (1,45,18)(2,43,13)(3,41,17)(4,39,12)(5,37,16)(6,44,11)(7,42,15)(8,40,10)(9,38,14)(19,66,52)(20,64,47)(21,71,51)(22,69,46)(23,67,50)(24,65,54)(25,72,49)(26,70,53)(27,68,48)(28,81,58)(29,79,62)(30,77,57)(31,75,61)(32,73,56)(33,80,60)(34,78,55)(35,76,59)(36,74,63), (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)>;
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), (1,80,26)(2,81,27)(3,73,19)(4,74,20)(5,75,21)(6,76,22)(7,77,23)(8,78,24)(9,79,25)(10,34,54)(11,35,46)(12,36,47)(13,28,48)(14,29,49)(15,30,50)(16,31,51)(17,32,52)(18,33,53)(37,61,71)(38,62,72)(39,63,64)(40,55,65)(41,56,66)(42,57,67)(43,58,68)(44,59,69)(45,60,70), (1,7,4)(2,10,71)(3,44,29)(5,13,65)(6,38,32)(8,16,68)(9,41,35)(11,25,66)(12,53,30)(14,19,69)(15,47,33)(17,22,72)(18,50,36)(20,26,23)(21,48,55)(24,51,58)(27,54,61)(28,40,75)(31,43,78)(34,37,81)(39,60,67)(42,63,70)(45,57,64)(46,79,56)(49,73,59)(52,76,62)(74,80,77), (1,45,18)(2,43,13)(3,41,17)(4,39,12)(5,37,16)(6,44,11)(7,42,15)(8,40,10)(9,38,14)(19,66,52)(20,64,47)(21,71,51)(22,69,46)(23,67,50)(24,65,54)(25,72,49)(26,70,53)(27,68,48)(28,81,58)(29,79,62)(30,77,57)(31,75,61)(32,73,56)(33,80,60)(34,78,55)(35,76,59)(36,74,63), (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) );
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)], [(1,80,26),(2,81,27),(3,73,19),(4,74,20),(5,75,21),(6,76,22),(7,77,23),(8,78,24),(9,79,25),(10,34,54),(11,35,46),(12,36,47),(13,28,48),(14,29,49),(15,30,50),(16,31,51),(17,32,52),(18,33,53),(37,61,71),(38,62,72),(39,63,64),(40,55,65),(41,56,66),(42,57,67),(43,58,68),(44,59,69),(45,60,70)], [(1,7,4),(2,10,71),(3,44,29),(5,13,65),(6,38,32),(8,16,68),(9,41,35),(11,25,66),(12,53,30),(14,19,69),(15,47,33),(17,22,72),(18,50,36),(20,26,23),(21,48,55),(24,51,58),(27,54,61),(28,40,75),(31,43,78),(34,37,81),(39,60,67),(42,63,70),(45,57,64),(46,79,56),(49,73,59),(52,76,62),(74,80,77)], [(1,45,18),(2,43,13),(3,41,17),(4,39,12),(5,37,16),(6,44,11),(7,42,15),(8,40,10),(9,38,14),(19,66,52),(20,64,47),(21,71,51),(22,69,46),(23,67,50),(24,65,54),(25,72,49),(26,70,53),(27,68,48),(28,81,58),(29,79,62),(30,77,57),(31,75,61),(32,73,56),(33,80,60),(34,78,55),(35,76,59),(36,74,63)], [(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)]])
C33.C32 is a maximal subgroup of
C32⋊C9⋊C6 C32⋊C9⋊S3 (C3×He3).S3
35 conjugacy classes
| class | 1 | 3A | ··· | 3H | 3I | ··· | 3P | 9A | ··· | 9R |
| order | 1 | 3 | ··· | 3 | 3 | ··· | 3 | 9 | ··· | 9 |
| size | 1 | 1 | ··· | 1 | 9 | ··· | 9 | 9 | ··· | 9 |
35 irreducible representations
| dim | 1 | 1 | 1 | 1 | 3 | 3 | 3 |
| type | + | ||||||
| image | C1 | C3 | C3 | C3 | He3 | C3≀C3 | He3.C3 |
| kernel | C33.C32 | C32⋊C9 | C3×He3 | C3×3- 1+2 | C32 | C3 | C3 |
| # reps | 1 | 2 | 2 | 4 | 2 | 18 | 6 |
Matrix representation of C33.C32 ►in GL6(𝔽19)
| 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 |
| 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 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 11 | 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 | 2 | 17 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 12 | 8 | 0 |
| 0 | 0 | 0 | 3 | 16 | 11 |
| 17 | 17 | 5 | 0 | 0 | 0 |
| 5 | 17 | 17 | 0 | 0 | 0 |
| 17 | 5 | 17 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 14 | 5 | 7 |
| 0 | 0 | 0 | 0 | 10 | 14 |
G:=sub<GL(6,GF(19))| [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],[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],[1,0,0,0,0,0,0,11,0,0,0,0,0,0,7,0,0,0,0,0,0,11,0,2,0,0,0,0,11,17,0,0,0,0,0,1],[0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,3,0,0,0,1,8,16,0,0,0,0,0,11],[17,5,17,0,0,0,17,17,5,0,0,0,5,17,17,0,0,0,0,0,0,0,14,0,0,0,0,0,5,10,0,0,0,1,7,14] >;
C33.C32 in GAP, Magma, Sage, TeX
C_3^3.C_3^2
% in TeX
G:=Group("C3^3.C3^2"); // GroupNames label
G:=SmallGroup(243,4);
// by ID
G=gap.SmallGroup(243,4);
# by ID
G:=PCGroup([5,-3,3,-3,-3,3,135,121,542,457]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^3=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
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