Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Sylow | C3 ⋊ #2546 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, prime radical, cosocle | C23×C3 | C24 |
| Fitting | C2×C3×C4⋊Q8 | C2 |
| hypercenter | C2×C4⋊Q8 | D3 |
| hypocenter | C3 | #2546 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C2×C4⋊Q8 | |
540 conjugacy classes of subgroups; only the 189 normal subgroups are shown.
This group has 540 conjugacy classes of subgroups — too many to draw. Only its 189 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) = 384 |
A minimal faithful permutation representation — 4 generators on 19 points:
(1,2,3)(12,15,16,14)(13,18,19,17);(2,3)(5,7)(8,9,10,11);(4,5,6,7)(12,14,16,15)(13,17,19,18);(2,3)(9,11)(12,13,16,19)(14,18,15,17)Polycyclic presentation:
2^7,3|a^2=e,b^2=eg,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 | #10692554, #10692594, #10692868, … |
| maximal normal subgroup of | #10692554, #10692594, #10692868, … |
| maximal quotient of | #10717464, #10717603, #10718357, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 15815);