Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| Sylow | C3 ⋊ #2309 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator | C22×C3 | C23×C4 |
| Fitting, prime radical | C23×C3×D4 | C2 |
| socle, cosocle | C23×C3 | C24 |
| hypercenter | C23×D4 | D3 |
| hypocenter | C3 | #2309 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C23×D4 | |
1104 conjugacy classes of subgroups; only the 309 normal subgroups are shown.
This group has 1104 conjugacy classes of subgroups — too many to draw. Only its 309 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 19 points:
(1,2,3)(13,17)(16,18);(9,11)(14,19)(16,18);(12,14,15,19)(13,16,17,18);(2,3)(4,5,6,7)(8,9)(10,11)(12,13)(14,18)(15,17)(16,19)Polycyclic presentation:
2^7,3|a^2=g,b^2=e,b^a=be,c^a=cf,d^a=de,h^a=h^2,c^b=ce84 conjugacy classes — view the full character table.
| maximal subgroup of | #10688690, #10688693, #10693094, … |
| maximal normal subgroup of | #10688690, #10688693, #10693094, … |
| maximal quotient of | #10716428, #10716496, #10716577, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 16620);