Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| Fitting | C3×#529 ⋊ C2 |
| Sylow | C3 ⋊ #3212 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C22×#67 |
| commutator, cosocle | C2×C3×C4 | C24 |
| Frattini | C2×C4 | C23×D3 |
| Fitting | C3×#529 | C2 |
| socle | C22×C3 | C22×D4 |
| hypercenter | #529 | D3 |
| hypocenter | C3 | #3212 |
| LCS | G ▷ C2×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #529 | |
400 conjugacy classes of subgroups; only the 121 normal subgroups are shown.
This group has 400 conjugacy classes of subgroups — too many to draw. Only its 121 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)(12,15,16,17)(13,20,18,21)(14,24,23,25)(19,22,26,27);(2,3)(6,11)(8,10)(12,14)(13,19)(15,24)(16,23)(17,25)(18,26)(20,22)(21,27);(4,6,7,11)(5,8,9,10)(13,18)(14,23)(20,21)(24,25);(4,5)(6,10)(7,9)(8,11)(12,13)(14,22)(15,21)(16,18)(17,20)(19,24)(23,27)(25,26)Polycyclic presentation:
2^7,3|b^2=eg,d^2=g,f^2=g,b^a=be,c^a=cf,d^a=de,f^a=fg,c^b=ce,f^c=fg,h^c=h^260 conjugacy classes — view the full character table.
| maximal subgroup of | #10772691, #10772697, #10772721, … |
| maximal normal subgroup of | #10772691, #10772697, #10772721, … |
| maximal quotient of | #10804515, #10804889, #10805730, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 15721);