Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C3 ⋊ #2577 |
| 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 | #2577 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 | |
412 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)(13,17)(16,18)(20,22,23,27)(21,25,24,26);(6,10)(8,11)(13,17)(14,19);(4,6,7,10)(5,8,9,11)(12,14)(13,16)(15,19)(17,18)(21,24)(25,26);(2,3)(4,5,7,9)(6,8,10,11)(12,13,15,17)(14,18,19,16)(20,21)(22,26)(23,24)(25,27)Polycyclic presentation:
2^7,3|a^2=g,b^2=f,d^2=efg,b^a=be,c^a=cfg,d^a=de,h^a=h^2,c^b=cg78 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#2577, C2×#73545, #10965673, … |
| maximal normal subgroup of | D3×#2577, C2×#73545, #10965673, … |
| maximal quotient of | #10717551, #10717826, #10718195, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 6121);