Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| Fitting | C3×#418 ⋊ C2 |
| Sylow | C3 ⋊ #3008 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×D3×D4 |
| commutator, cosocle | C2×C3×C4 | C24 |
| Frattini | C2×C4 | C23×D3 |
| Fitting | C3×#418 | C2 |
| socle | C22×C3 | C22×D4 |
| hypercenter | #418 | D3 |
| hypocenter | C3 | #3008 |
| LCS | G ▷ C2×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #418 | |
784 conjugacy classes of subgroups; only the 149 normal subgroups are shown.
This group has 784 conjugacy classes of subgroups — too many to draw. Only its 149 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)(10,17)(14,21)(18,23)(20,22);(2,3)(5,7)(8,10)(9,14)(11,18)(12,17)(13,20)(15,22)(16,21)(19,23);(9,13)(11,19)(14,20)(15,16)(18,23)(21,22);(2,3)(4,5)(6,7)(8,9)(10,14)(11,13)(12,16)(15,19)(17,21)(18,20)(22,23)Polycyclic presentation:
2^7,3|e^2=g,b^a=be,c^a=cf,e^a=eg,h^a=h^2,e^b=eg,d^c=dg,h^c=h^254 conjugacy classes — view the full character table.
| maximal subgroup of | #10826704, #10826710, #10832571, … |
| maximal normal subgroup of | #10826704, #10826710, #10832571, … |
| maximal quotient of | #10768350, #10768438, #10768525, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 11504);