Finite Group DatabasebetaFinite Group Databasebeta| order | 243 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6856 |
| automorphism | Aut(G) = C32×#66432 (order 2916) · Inn(G) = H(3,𝔽3) · Out(G) = C2×C32×D3 |
| homology | Abelianization (H1) = C32 · Schur multiplier (H2) = C3 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C32 | H(3,𝔽3) |
| commutator, Frattini, prime radical, cosocle | C33 | C32 |
| LCS | G ▷ C33 ▷ C32 ▷ C1 | |
| UCS | C1 ◁ C32 ◁ C33 ◁ G | |
| LECS | G ▷ C33 ▷ C32 ▷ C1 | |
25 conjugacy classes of subgroups.
| minimal degree | μ(G) = 54 |
| transitive degree | μt(G) = 81 |
A minimal faithful permutation representation — 2 generators on 54 points:
(1,2,5)(3,9,13)(4,8,14)(6,15,11)(7,18,12)(10,20,23)(16,26,27)(17,25,22)(19,24,21)(28,29,32,31,34,41,39,45,51)(30,35,46,38,33,42,37,44,40)(36,43,50,48,53,54,49,47,52);(1,3,10,4,11,21,12,22,27)(2,6,16,8,17,20,7,9,19)(5,13,23,14,15,24,18,25,26)(28,30,36)(29,33,43)(31,38,48)(32,40,50)(34,44,53)(35,47,45)(37,49,39)(41,46,54)(42,52,51)Polycyclic presentation:
3^5|a^3=e,b^3=d,b^a=bc,c^a=cd,c^b=ce35 conjugacy classes — view the full character table.
| derived subgroup of | #92203, #11757326, #92165⋊C2, … |
| Fitting subgroup of | #92165, #92166, #92203, … |
| maximal subgroup of | #92165, #92166, #92203, … |
| maximal normal subgroup of | #92165, #92166, #92203, … |
| maximal quotient of | C2×#6861, #10624450, #10624451, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(243, 8);