Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| Sylow | C3 ⋊ #2574 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, cosocle | C23×C3 | C24 |
| Fitting | C3×C4×C22⋊C4 | C2 |
| prime radical | C24×C3 | C23 |
| hypercenter | C4×C22⋊C4 | D3 |
| hypocenter | C3 | #2574 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 | |
436 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3)(13,16)(17,19)(20,22,23,26)(21,25,24,27);(6,10)(8,11)(12,14,15,18)(13,17,16,19)(21,24)(25,27);(4,6,7,10)(5,8,9,11)(13,16)(14,18);(2,3)(4,5,7,9)(6,8,10,11)(12,13)(14,17)(15,16)(18,19)(20,21)(22,25)(23,24)(26,27)Polycyclic presentation:
2^7,3|a^2=e,b^2=e,c^2=eg,d^2=f,b^a=beg,c^a=cf,d^a=deg,h^a=h^2,c^b=cg78 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#2574, C2×#73542, #10965541, … |
| maximal normal subgroup of | D3×#2574, C2×#73542, #10965541, … |
| maximal quotient of | #10717513, #10717616, #10718165, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 6118);