Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| Sylow | C3 ⋊ #2769 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, prime radical, cosocle | C23×C3 | C24 |
| Fitting | C3×#382 | C2 |
| hypercenter | #382 | D3 |
| hypocenter | C3 | #2769 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #382 | |
344 conjugacy classes of subgroups; only the 121 normal subgroups are shown.
This group has 344 conjugacy classes of subgroups — too many to draw. Only its 121 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) = 384 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2)(12,15,16,14)(13,18,17,19)(21,26)(25,27);(2,3)(6,10)(8,11)(12,14,16,15)(13,18,17,19)(20,23,24,22)(21,25,26,27);(4,6)(5,8)(7,10)(9,11)(13,17)(14,15)(20,22,24,23)(21,25,26,27);(4,5,7,9)(6,8,10,11)(12,13,16,17)(14,18,15,19)(20,21)(22,25)(23,27)(24,26)Polycyclic presentation:
2^7,3|a^2=g,b^2=f,c^2=ef,d^2=e,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,h^c=h^2,h^d=h^260 conjugacy classes — view the full character table.
| maximal subgroup of | #10707558, #10707562, #10708144, … |
| maximal normal subgroup of | #10707558, #10707562, #10708144, … |
| maximal quotient of | #10722896, #10723002, #10723022, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 6785);