Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| Fitting | C3×#532 ⋊ C2 |
| Sylow | C3 ⋊ #3153 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C22×#67 |
| commutator, cosocle | C2×C3×C4 | C24 |
| Frattini | C2×C4 | C23×D3 |
| Fitting | C3×#532 | C2 |
| socle | C22×C3 | C22×D4 |
| hypercenter | #532 | D3 |
| hypocenter | C3 | #3153 |
| LCS | G ▷ C2×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #532 | |
572 conjugacy classes of subgroups; only the 149 normal subgroups are shown.
This group has 572 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)(13,21)(16,22)(18,23);(4,5,6,7)(9,14)(13,21)(15,19)(16,22);(2,3)(8,10,11,18,12,17,20,23)(9,13,15,16,14,21,19,22);(2,3)(5,7)(8,9)(10,16)(11,19)(12,14)(13,18)(15,20)(17,22)(21,23)Polycyclic presentation:
2^7,3|b^2=e,c^2=fg,e^2=g,b^a=be,c^a=cf,e^a=eg,h^a=h^2,d^b=dg,h^b=h^260 conjugacy classes — view the full character table.
| maximal subgroup of | #10692813, #10696060, #10696062, … |
| maximal normal subgroup of | #10692813, #10696060, #10696062, … |
| maximal quotient of | #10783054, #10783065, #10783098, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 15865);