Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C214 |
No aliases for this group.
| Fitting | C2×C3×H(5,𝔽2) ⋊ C2 |
| Sylow | C3 ⋊ H(7,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C25×D3 |
| commutator, socle, cosocle | C2×C3 | C26 |
| Fitting | C2×C3×H(5,𝔽2) | C2 |
| hypercenter | C2×H(5,𝔽2) | D3 |
| hypocenter | C3 | H(7,𝔽2) |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2 ◁ C2×H(5,𝔽2) | |
5992 conjugacy classes of subgroups; only the 3201 normal subgroups are shown.
This group has 5992 conjugacy classes of subgroups — too many to draw. Only its 3201 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) = 48 |
A minimal faithful permutation representation — 6 generators on 19 points:
(1,2,3)(5,11)(9,12)(10,17)(16,18);(6,14)(9,12)(13,19)(16,18);(7,15)(10,17)(13,19)(16,18);(4,7)(5,10)(6,13)(8,15)(9,16)(11,17)(12,18)(14,19);(4,6)(5,9)(7,13)(8,14)(10,16)(11,12)(15,19)(17,18);(2,3)(4,5)(6,12)(7,10)(8,11)(9,14)(13,18)(15,17)(16,19)Polycyclic presentation:
2^7,3|b^a=bg,f^a=fg,h^a=h^2,e^b=eg,d^c=dg99 conjugacy classes — view the full character table.
| maximal subgroup of | #10943929, #10943933, #10943943, … |
| maximal normal subgroup of | #10943929, #10943933, #10943943, … |
| maximal quotient of | #11701803, #11701804, #11701806, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 20144);