Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C3×#407 ⋊ C2 |
| Sylow | C3 ⋊ #3197 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C22×D12 |
| commutator, cosocle | C2×C3×C4 | C24 |
| Frattini | C2×C4 | C23×D3 |
| Fitting | C3×#407 | C2 |
| socle | C22×C3 | C22×D4 |
| hypercenter | #407 | D3 |
| hypocenter | C3 | #3197 |
| LCS | G ▷ C2×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ #407 | |
418 conjugacy classes of subgroups; only the 117 normal subgroups are shown.
This group has 418 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) = 96 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3)(5,9)(8,10)(12,15,17,16)(13,19,20,24)(14,23,22,25)(18,27,26,21);(2,3)(6,11)(8,10)(12,14)(13,18)(15,23)(16,25)(17,22)(19,27)(20,26)(21,24);(4,6)(5,8)(7,11)(9,10)(14,22)(18,26)(21,27)(23,25);(2,3)(4,5)(6,10)(7,9)(8,11)(12,13)(14,21)(15,19)(16,24)(17,20)(18,23)(22,27)(25,26)Polycyclic presentation:
2^7,3|d^2=g,f^2=g,b^a=be,c^a=cf,d^a=de,f^a=fg,h^a=h^2,c^b=ceg,f^c=fg,h^c=h^254 conjugacy classes — view the full character table.
| maximal subgroup of | #10707361, #10707365, #10708521, … |
| maximal normal subgroup of | #10707361, #10707365, #10708521, … |
| maximal quotient of | #10804255, #10805474, #10806418, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 11220);