Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C3×#382 ⋊ C2 |
| Sylow | C3 ⋊ #2621 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, cosocle | C23×C3 | C24 |
| Fitting | C3×#382 | C2 |
| prime radical | C23×D3 | C23 |
| hypercenter | #382 | D3 |
| hypocenter | C3 | #2621 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #382 | |
528 conjugacy classes of subgroups; only the 157 normal subgroups are shown.
This group has 528 conjugacy classes of subgroups — too many to draw. Only its 157 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)(12,15,16,23)(13,19,17,25)(14,20,21,22)(18,26,24,27);(2,3)(5,7)(9,11)(12,14,16,21)(13,18,17,24)(15,22,23,20)(19,27,25,26);(2,3)(8,9)(10,11)(13,17)(18,24)(19,25)(26,27);(4,5,6,7)(12,13)(14,18)(15,19)(16,17)(20,26)(21,24)(22,27)(23,25)Polycyclic presentation:
2^7,3|a^2=f,c^2=e,d^2=e,b^a=be,c^a=cf,c^b=cg,h^b=h^2,d^c=de,h^c=h^260 conjugacy classes — view the full character table.
| maximal subgroup of | #10691209, #10691214, #10769227, … |
| maximal normal subgroup of | #10691209, #10691214, #10769227, … |
| maximal quotient of | #10721565, #10721592, #10721722, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 6769);