Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Sylow | C3 ⋊ #2677 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, cosocle | C23×C3 | C24 |
| Fitting | C3×#399 | C2 |
| prime radical | C24×C3 | C23 |
| hypercenter | #399 | D3 |
| hypocenter | C3 | #2677 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #399 | |
544 conjugacy classes of subgroups; only the 149 normal subgroups are shown.
This group has 544 conjugacy classes of subgroups — too many to draw. Only its 149 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) = 192 |
A minimal faithful permutation representation — 4 generators on 19 points:
(1,2)(5,7)(12,15,16,17)(13,14,18,19);(5,7)(9,11);(2,3)(8,9)(10,11)(12,14,16,19)(13,17,18,15);(2,3)(4,5)(6,7)(12,13,16,18)(14,15,19,17)Polycyclic presentation:
2^7,3|a^2=e,b^2=e,d^2=e,b^a=be,c^a=cf,d^a=def,h^a=h^2,c^b=cg,d^b=de,h^b=h^2,h^d=h^254 conjugacy classes — view the full character table.
| maximal subgroup of | #10707730, #10707734, #10708214, … |
| maximal normal subgroup of | #10707730, #10707734, #10708214, … |
| maximal quotient of | #10722543, #10723121, #10723161, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 10637);