Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C3 ⋊ #2771 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, cosocle | C23×C3 | C24 |
| Fitting | C3×#399 | C2 |
| prime radical | C24×C3 | C23 |
| hypercenter | #399 | D3 |
| hypocenter | C3 | #2771 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #399 | |
392 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 392 conjugacy classes of subgroups — too many to draw. Only its 125 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3)(14,19)(16,18)(21,26)(25,27);(2,3)(6,10)(8,11)(13,17)(14,19)(20,23,24,22)(21,25,26,27);(2,3)(4,6)(5,8)(7,10)(9,11)(12,14)(13,16)(15,19)(17,18)(20,22,24,23)(21,25,26,27);(2,3)(4,5,7,9)(6,8,10,11)(12,13)(14,18)(15,17)(16,19)(20,21,24,26)(22,25,23,27)Polycyclic presentation:
2^7,3|a^2=f,b^2=efg,c^2=efg,b^a=be,c^a=cfg,d^a=defg,h^a=h^2,c^b=cg,d^b=de,h^b=h^2,h^c=h^254 conjugacy classes — view the full character table.
| maximal subgroup of | #10707813, #10707817, #10708297, … |
| maximal normal subgroup of | #10707813, #10707817, #10708297, … |
| maximal quotient of | #10722862, #10723560, #10723884, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 10657);