Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C3×#459 ⋊ C2 |
| Sylow | C3 ⋊ #3284 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×D3×D4 |
| commutator, cosocle | C2×C3×C4 | C24 |
| Frattini | C2×C4 | C23×D3 |
| Fitting | C3×#459 | C2 |
| socle | C22×C3 | C22×D4 |
| hypercenter | #459 | D3 |
| hypocenter | C3 | #3284 |
| LCS | G ▷ C2×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #459 | |
494 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 494 conjugacy classes of subgroups — too many to draw. Only its 125 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) = 192 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2,3)(8,9)(10,19)(11,15)(12,21)(13,16)(14,22)(17,20)(18,23);(2,3)(5,7)(8,11,13,12)(9,15,16,21)(10,18,20,22)(14,19,23,17);(2,3)(8,10,13,20)(9,14,16,23)(11,18,12,22)(15,19,21,17);(2,3)(4,5)(6,7)(8,9,13,16)(10,17,20,19)(11,15,12,21)(14,22,23,18)Polycyclic presentation:
2^7,3|a^2=g,b^2=g,c^2=g,e^2=g,b^a=be,c^a=cf,d^a=dg,e^a=eg,h^a=h^2,d^b=de,e^b=eg,h^b=h^2,h^c=h^2,e^d=eg54 conjugacy classes — view the full character table.
| maximal subgroup of | #10784592, #10784600, #10784607, … |
| maximal normal subgroup of | #10784592, #10784600, #10784607, … |
| maximal quotient of | #10785971, #10785986, #10786001, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 12927);