Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| Fitting | C3×#386 ⋊ C2 |
| Sylow | C3 ⋊ #2366 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator | C22×C3 | C23×C4 |
| Fitting | C3×#386 | C2 |
| socle, cosocle | C23×C3 | C24 |
| prime radical | C2×#940 | C2 |
| hypercenter | #386 | D3 |
| hypocenter | C3 | #2366 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #386 | |
832 conjugacy classes of subgroups; only the 209 normal subgroups are shown.
This group has 832 conjugacy classes of subgroups — too many to draw. Only its 209 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) = 192 |
A minimal faithful permutation representation — 4 generators on 15 points:
(1,2)(12,14,15,13);(2,3)(8,9,10,11);(4,6,7,5)(13,14);(4,5,7,6)(9,11)(12,13,15,14)Polycyclic presentation:
2^7,3|a^2=g,b^2=eg,c^2=f,d^2=e,b^a=be,c^a=cf,d^b=de,h^c=h^2,h^d=h^278 conjugacy classes — view the full character table.
| maximal subgroup of | #10711979, #10711982, #10712486, … |
| maximal normal subgroup of | #10711979, #10711982, #10712486, … |
| maximal quotient of | #10716625, #10716750, #10716757, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 7255);