Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C3×#576 ⋊ C2 |
| Sylow | C3 ⋊ #3416 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C22×#213 |
| commutator, cosocle | C3×C8 | C24 |
| Frattini | C8 | C23×D3 |
| Fitting | C3×#576 | C2 |
| socle | C2×C3 | C22×D8 |
| hypercenter | #576 | D3 |
| hypocenter | C3 | #3416 |
| LCS | G ▷ C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2 ◁ #41 ◁ #131 ◁ #576 | |
360 conjugacy classes of subgroups; only the 129 normal subgroups are shown.
This group has 360 conjugacy classes of subgroups — too many to draw. Only its 129 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 35 points:
(1,2,3)(6,15)(11,25)(16,29)(17,30)(22,33)(26,34)(27,32)(31,35);(4,6)(5,11)(7,16)(8,17)(9,15)(10,22)(12,26)(13,27)(14,25)(18,31)(19,30)(20,29)(21,32)(23,34)(24,33)(28,35);(5,10)(7,18)(8,19)(11,22)(12,13)(14,24)(16,31)(17,30)(20,28)(21,23)(25,33)(26,27)(29,35)(32,34);(2,3)(4,5,9,14)(6,11,15,25)(7,10,20,24)(8,21,19,13)(12,28,23,18)(16,22,29,33)(17,32,30,27)(26,35,34,31)Polycyclic presentation:
2^7,3|a^2=g,e^2=fg,f^2=g,b^a=be,e^a=ef,f^a=fg,h^a=h^2,e^b=ef,f^b=fg,d^c=dg54 conjugacy classes — view the full character table.
| maximal subgroup of | #10951929, #10951931, #10951944, … |
| maximal normal subgroup of | #10951929, #10951931, #10951944, … |
| maximal quotient of | #10948099, #10948103, #10948234, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 16584);