Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| Fitting | C3×#451 ⋊ C2 |
| Sylow | C3 ⋊ #3002 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×D3×D4 |
| commutator, cosocle | C2×C3×C4 | C24 |
| Frattini | C2×C4 | C23×D3 |
| Fitting | C3×#451 | C2 |
| socle | C22×C3 | C22×D4 |
| hypercenter | #451 | D3 |
| hypocenter | C3 | #3002 |
| LCS | G ▷ C2×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #451 | |
764 conjugacy classes of subgroups; only the 141 normal subgroups are shown.
This group has 764 conjugacy classes of subgroups — too many to draw. Only its 141 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)(8,11,13,12)(9,15,16,20)(10,18,19,21)(14,22,23,17);(5,7);(2,3)(8,10)(9,14)(11,18)(12,21)(13,19)(15,22)(16,23)(17,20);(2,3)(4,5)(6,7)(8,9)(10,17)(11,15)(12,20)(13,16)(14,18)(19,22)(21,23)Polycyclic presentation:
2^7,3|d^2=g,e^2=g,b^a=be,c^a=cf,e^a=eg,h^a=h^2,e^b=eg,h^b=h^2,h^d=h^257 conjugacy classes — view the full character table.
| maximal subgroup of | #10767296, #10767298, #10767398, … |
| maximal normal subgroup of | #10767296, #10767298, #10767398, … |
| maximal quotient of | #10767298, #10767349, #10767400, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 12758);