Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Sylow | C3 ⋊ #1713 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | #874 |
| commutator, cosocle | C3×C42 | C23 |
| Frattini | C42 | C22×D3 |
| Fitting | C3×C82 | C2 |
| socle, prime radical | C22×C3 | #127 |
| hypercenter | C82 | D3 |
| hypocenter | C3 | #1713 |
| LCS | G ▷ C3×C42 ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C82 | |
192 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,3)(4,5,9,11,7,10,6,8);(12,14,21,25,16,23,15,22)(13,17,24,27,19,26,18,20);(2,3)(5,8)(6,9)(10,11)(12,13,16,19)(14,20,23,27)(15,24,21,18)(17,25,26,22)Polycyclic presentation:
2^7,3|a^2=f,b^2=df,c^2=eg,d^2=f,e^2=g,b^a=bd,c^a=ce,d^a=df,e^a=eg,h^a=h^2102 conjugacy classes — view the full character table.
| maximal subgroup of | #10681919, #10681921, #10681922, … |
| maximal normal subgroup of | #10681919, #10681921, #10681922, … |
| maximal quotient of | #10688968, #10694946, #10694954, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 641);