p-group, metabelian, nilpotent (class 2), monomial
Aliases: C32⋊C27, C9.3He3, C33.1C9, C9.63- 1+2, (C3×C27)⋊1C3, (C3×C9).2C9, C3.1(C3×C27), C3.2(C27⋊C3), (C32×C9).2C3, C3.2(C32⋊C9), C32.14(C3×C9), (C3×C9).31C32, SmallGroup(243,12)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C32⋊C27
G = < a,b,c | a3=b3=c27=1, ab=ba, cac-1=ab-1, bc=cb >
Smallest permutation representation of C32⋊C27
►On 81 points
Generators in S81
(1 10 19)(2 72 33)(3 52 55)(4 13 22)(5 75 36)(6 28 58)(7 16 25)(8 78 39)(9 31 61)(11 81 42)(12 34 64)(14 57 45)(15 37 67)(17 60 48)(18 40 70)(20 63 51)(21 43 73)(23 66 54)(24 46 76)(26 69 30)(27 49 79)(29 38 47)(32 41 50)(35 44 53)(56 65 74)(59 68 77)(62 71 80)
(1 41 62)(2 42 63)(3 43 64)(4 44 65)(5 45 66)(6 46 67)(7 47 68)(8 48 69)(9 49 70)(10 50 71)(11 51 72)(12 52 73)(13 53 74)(14 54 75)(15 28 76)(16 29 77)(17 30 78)(18 31 79)(19 32 80)(20 33 81)(21 34 55)(22 35 56)(23 36 57)(24 37 58)(25 38 59)(26 39 60)(27 40 61)
(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)| (1,10,19)(2,72,33)(3,52,55)(4,13,22)(5,75,36)(6,28,58)(7,16,25)(8,78,39)(9,31,61)(11,81,42)(12,34,64)(14,57,45)(15,37,67)(17,60,48)(18,40,70)(20,63,51)(21,43,73)(23,66,54)(24,46,76)(26,69,30)(27,49,79)(29,38,47)(32,41,50)(35,44,53)(56,65,74)(59,68,77)(62,71,80), (1,41,62)(2,42,63)(3,43,64)(4,44,65)(5,45,66)(6,46,67)(7,47,68)(8,48,69)(9,49,70)(10,50,71)(11,51,72)(12,52,73)(13,53,74)(14,54,75)(15,28,76)(16,29,77)(17,30,78)(18,31,79)(19,32,80)(20,33,81)(21,34,55)(22,35,56)(23,36,57)(24,37,58)(25,38,59)(26,39,60)(27,40,61), (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( (1,10,19)(2,72,33)(3,52,55)(4,13,22)(5,75,36)(6,28,58)(7,16,25)(8,78,39)(9,31,61)(11,81,42)(12,34,64)(14,57,45)(15,37,67)(17,60,48)(18,40,70)(20,63,51)(21,43,73)(23,66,54)(24,46,76)(26,69,30)(27,49,79)(29,38,47)(32,41,50)(35,44,53)(56,65,74)(59,68,77)(62,71,80), (1,41,62)(2,42,63)(3,43,64)(4,44,65)(5,45,66)(6,46,67)(7,47,68)(8,48,69)(9,49,70)(10,50,71)(11,51,72)(12,52,73)(13,53,74)(14,54,75)(15,28,76)(16,29,77)(17,30,78)(18,31,79)(19,32,80)(20,33,81)(21,34,55)(22,35,56)(23,36,57)(24,37,58)(25,38,59)(26,39,60)(27,40,61), (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([[(1,10,19),(2,72,33),(3,52,55),(4,13,22),(5,75,36),(6,28,58),(7,16,25),(8,78,39),(9,31,61),(11,81,42),(12,34,64),(14,57,45),(15,37,67),(17,60,48),(18,40,70),(20,63,51),(21,43,73),(23,66,54),(24,46,76),(26,69,30),(27,49,79),(29,38,47),(32,41,50),(35,44,53),(56,65,74),(59,68,77),(62,71,80)], [(1,41,62),(2,42,63),(3,43,64),(4,44,65),(5,45,66),(6,46,67),(7,47,68),(8,48,69),(9,49,70),(10,50,71),(11,51,72),(12,52,73),(13,53,74),(14,54,75),(15,28,76),(16,29,77),(17,30,78),(18,31,79),(19,32,80),(20,33,81),(21,34,55),(22,35,56),(23,36,57),(24,37,58),(25,38,59),(26,39,60),(27,40,61)], [(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)]])
C32⋊C27 is a maximal subgroup of
C32⋊C54 C32⋊D27 C32⋊2D27
99 conjugacy classes
| class | 1 | 3A | ··· | 3H | 3I | ··· | 3N | 9A | ··· | 9R | 9S | ··· | 9AD | 27A | ··· | 27BB |
| order | 1 | 3 | ··· | 3 | 3 | ··· | 3 | 9 | ··· | 9 | 9 | ··· | 9 | 27 | ··· | 27 |
| size | 1 | 1 | ··· | 1 | 3 | ··· | 3 | 1 | ··· | 1 | 3 | ··· | 3 | 3 | ··· | 3 |
99 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 |
| type | + | ||||||||
| image | C1 | C3 | C3 | C9 | C9 | C27 | He3 | 3- 1+2 | C27⋊C3 |
| kernel | C32⋊C27 | C3×C27 | C32×C9 | C3×C9 | C33 | C32 | C9 | C9 | C3 |
| # reps | 1 | 6 | 2 | 12 | 6 | 54 | 2 | 4 | 12 |
Matrix representation of C32⋊C27 ►in GL4(𝔽109) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 45 | 0 |
| 0 | 0 | 0 | 63 |
| 1 | 0 | 0 | 0 |
| 0 | 45 | 0 | 0 |
| 0 | 0 | 45 | 0 |
| 0 | 0 | 0 | 45 |
| 25 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 63 | 0 | 0 |
G:=sub<GL(4,GF(109))| [1,0,0,0,0,1,0,0,0,0,45,0,0,0,0,63],[1,0,0,0,0,45,0,0,0,0,45,0,0,0,0,45],[25,0,0,0,0,0,0,63,0,1,0,0,0,0,1,0] >;
C32⋊C27 in GAP, Magma, Sage, TeX
C_3^2\rtimes C_{27} % in TeX
G:=Group("C3^2:C27"); // GroupNames label
G:=SmallGroup(243,12);
// by ID
G=gap.SmallGroup(243,12);
# by ID
G:=PCGroup([5,-3,3,-3,3,-3,135,121,78]);
// Polycyclic
G:=Group<a,b,c|a^3=b^3=c^27=1,a*b=b*a,c*a*c^-1=a*b^-1,b*c=c*b>;
// generators/relations
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