Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| Sylow | C3 ⋊ #2501 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, cosocle | C23×C3 | C24 |
| Fitting, prime radical | C2×C3×#121 | C2 |
| hypercenter | C2×#121 | D3 |
| hypocenter | C3 | #2501 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C2×#121 | |
708 conjugacy classes of subgroups; only the 181 normal subgroups are shown.
This group has 708 conjugacy classes of subgroups — too many to draw. Only its 181 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:
(1,2,3)(8,10,11,9);(2,3)(4,5,6,7)(14,18)(16,19);(8,9,11,10)(12,14)(13,16)(15,18)(17,19);(2,3)(5,7)(9,10)(12,13,15,17)(14,16,18,19)Polycyclic presentation:
2^7,3|a^2=g,b^2=e,c^2=f,d^2=e,b^a=be,c^a=cf,d^a=de,h^a=h^2,c^b=cg,h^c=h^266 conjugacy classes — view the full character table.
| maximal subgroup of | #10692928, #10695647, #10695651, … |
| maximal normal subgroup of | #10692928, #10695647, #10695651, … |
| maximal quotient of | #10717173, #10717419, #10718311, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 15240);