Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| Fitting | C3×#536 ⋊ C2 |
| Sylow | C3 ⋊ #3221 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C22×#67 |
| commutator, cosocle | C2×C3×C4 | C24 |
| Frattini | C2×C4 | C23×D3 |
| Fitting | C3×#536 | C2 |
| socle | C22×C3 | C22×D4 |
| hypercenter | #536 | D3 |
| hypocenter | C3 | #3221 |
| LCS | G ▷ C2×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ #117 ◁ #536 | |
388 conjugacy classes of subgroups; only the 117 normal subgroups are shown.
This group has 388 conjugacy classes of subgroups — too many to draw. Only its 117 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3)(5,9)(8,10)(13,18)(17,25)(19,26)(24,27);(6,11)(8,10)(12,14,16,21)(13,17,18,25)(15,23,22,20)(19,27,26,24);(4,6,7,11)(5,8,9,10)(12,13)(14,17)(15,19)(16,18)(20,24)(21,25)(22,26)(23,27);(2,3)(4,5)(6,10)(7,9)(8,11)(14,20)(15,22)(17,24)(19,26)(21,23)(25,27)Polycyclic presentation:
2^7,3|b^2=e,c^2=g,f^2=g,b^a=be,c^a=cf,d^a=de,f^a=fg,h^a=h^2,c^b=ce,d^b=dg,f^c=fg48 conjugacy classes — view the full character table.
| maximal subgroup of | #10715576, #10715579, #10715601, … |
| maximal normal subgroup of | #10715576, #10715579, #10715601, … |
| maximal quotient of | #10804572, #10804974, #10805777, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 15985);