Finite Group DatabasebetaFinite Group Databasebeta| order | 243 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #6878 |
| automorphism | Aut(G) = #72277556 (order 8748) · Inn(G) = #732 · Out(G) = C3×D32 |
| homology | Abelianization (H1) = C32 · Schur multiplier (H2) = C3×C9 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C3 | #732 |
| commutator, Frattini, cosocle | C3×C9 | C32 |
| LCS | G ▷ C3×C9 ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3 ◁ C32 ◁ C3×C9 ◁ G | |
| LECS | G ▷ C3×C9 ▷ C32 ▷ C3 ▷ C1 | |
24 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 27 |
A minimal faithful permutation representation — 2 generators on 27 points:
(1,2,5)(3,7,12)(4,8,15)(6,16,21)(9,19,23)(10,17,13)(11,20,14)(18,25,27)(22,26,24);(2,6,17,8,18,26,20,19,7)(5,12,23,14,24,25,15,13,16)Polycyclic presentation:
3^5|b^3=d^2e,c^3=e^2,b^a=bc,c^a=cd,d^a=de35 conjugacy classes — view the full character table.
| derived subgroup of | #92216, #11757339, #92191⋊C2, … |
| Fitting subgroup of | #92191, #92192, #92216, … |
| maximal subgroup of | #92191, #92192, #92216, … |
| maximal normal subgroup of | #92191, #92192, #92216, … |
| maximal quotient of | C2×#6879, #10624480, #10624489, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(243, 26);