Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| Fitting | C2×C3×#95 ⋊ C2 |
| Sylow | C3 ⋊ #2430 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, cosocle | C23×C3 | C24 |
| Fitting | C2×C3×#95 | C2 |
| hypercenter | C2×#95 | D3 |
| hypocenter | C3 | #2430 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C2×#95 | |
914 conjugacy classes of subgroups; only the 173 normal subgroups are shown.
This group has 914 conjugacy classes of subgroups — too many to draw. Only its 173 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)(13,17)(16,18);(5,7)(8,9,10,11);(4,5)(6,7)(12,14,15,19)(13,16,17,18);(2,3)(9,11)(12,13)(14,18)(15,17)(16,19)Polycyclic presentation:
2^7,3|b^2=e,c^2=f,b^a=be,c^a=cf,d^a=de,h^a=h^2,c^b=cg78 conjugacy classes — view the full character table.
| maximal subgroup of | #10768675, #10768679, #10768690, … |
| maximal normal subgroup of | #10768675, #10768679, #10768690, … |
| maximal quotient of | #10717156, #10717872, #10718047, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5952);