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 ⋊ #2358 |
| 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×#937 | C2 |
| hypercenter | #386 | D3 |
| hypocenter | C3 | #2358 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #386 | |
928 conjugacy classes of subgroups; only the 209 normal subgroups are shown.
This group has 928 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) = 96 |
A minimal faithful permutation representation — 4 generators on 15 points:
(1,2)(13,15);(2,3)(9,11);(4,6,7,5)(12,13)(14,15);(2,3)(4,5,7,6)(8,9)(10,11)(13,15)Polycyclic presentation:
2^7,3|a^2=g,b^2=g,b^a=be,c^a=cf,h^a=h^2,d^b=de,h^c=h^2,h^d=h^278 conjugacy classes — view the full character table.
| maximal subgroup of | #10711937, #10711941, #10712444, … |
| maximal normal subgroup of | #10711937, #10711941, #10712444, … |
| maximal quotient of | #10716588, #10716688, #10716690, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 7456);