Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C3×C4×D8 ⋊ C2 |
| Sylow | C3 ⋊ #3273 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×D3×D4 |
| commutator, cosocle | C2×C3×C4 | C24 |
| Frattini | C2×C4 | C23×D3 |
| Fitting | C3×C4×D8 | C2 |
| socle | C22×C3 | C22×D4 |
| hypercenter | C4×D8 | D3 |
| hypocenter | C3 | #3273 |
| LCS | G ▷ C2×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C4×D8 | |
520 conjugacy classes of subgroups; only the 129 normal subgroups are shown.
This group has 520 conjugacy classes of subgroups — too many to draw. Only its 129 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2,3)(10,18)(11,21)(13,23)(14,20)(16,22)(17,19);(2,3)(5,7)(10,17)(13,22)(16,23)(18,19);(8,10)(9,13)(11,18)(12,17)(14,23)(15,22)(16,20)(19,21);(2,3)(4,5)(6,7)(8,9,12,15)(10,16,17,23)(11,20,21,14)(13,18,22,19)Polycyclic presentation:
2^7,3|a^2=g,e^2=g,b^a=be,c^a=cf,e^a=eg,h^a=h^2,c^b=cg,d^b=de,e^b=eg,h^c=h^2,e^d=eg57 conjugacy classes — view the full character table.
| maximal subgroup of | #10708666, #10708670, #10708680, … |
| maximal normal subgroup of | #10708666, #10708670, #10708680, … |
| maximal quotient of | #10786464, #10786548, #10786664, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 12180);