Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| Fitting | C3×#394 ⋊ C2 |
| Sylow | C3 ⋊ #2408 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, cosocle | C23×C3 | C24 |
| Fitting | C3×#394 | C2 |
| hypercenter | #394 | D3 |
| hypocenter | C3 | #2408 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #394 | |
1232 conjugacy classes of subgroups; only the 171 normal subgroups are shown.
This group has 1232 conjugacy classes of subgroups — too many to draw. Only its 171 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 15 points:
(1,2)(12,14,15,13);(2,3)(5,7)(9,11);(8,9)(10,11)(12,13,15,14);(4,5)(6,7)(13,14)Polycyclic presentation:
2^7,3|b^2=e,d^2=e,b^a=be,c^a=cf,d^a=de,c^b=cg,h^c=h^2,h^d=h^260 conjugacy classes — view the full character table.
| maximal subgroup of | #10705768, #10706231, #10767579, … |
| maximal normal subgroup of | #10705768, #10706231, #10767579, … |
| maximal quotient of | #10717091, #10717401, #10717952, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 9471);