Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C3×#408 ⋊ C2 |
| Sylow | C3 ⋊ #2017 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | #940 |
| commutator, cosocle | C22×C3×C4 | C23 |
| Frattini | C22×C4 | C22×D3 |
| Fitting | C3×#408 | C2 |
| socle | C2×C3 | #392 |
| hypercenter | #408 | D3 |
| hypocenter | C3 | #2017 |
| LCS | G ▷ C22×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ #408 | |
350 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 350 conjugacy classes of subgroups — too many to draw. Only its 69 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 — 3 generators on 35 points:
(1,2,3)(4,7)(5,12)(6,16)(8,22)(9,23)(10,24)(11,26)(13,30)(14,31)(15,32)(17,21)(18,35)(19,34)(20,25)(27,29)(28,33);(2,3)(4,6)(5,11)(7,21)(8,18)(9,17)(10,19)(12,29)(13,27)(14,28)(15,25)(16,22)(20,26)(23,34)(24,35)(30,33)(31,32);(2,3)(4,5)(6,15)(7,20)(8,25)(9,13)(10,14)(11,17)(12,23)(16,33)(18,28)(19,27)(21,32)(22,31)(24,30)(26,34)(29,35)Polycyclic presentation:
2^7,3|d^2=g,e^2=g,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eg,h^a=h^2,c^b=cf,d^b=dg,f^b=fg,h^b=h^2,e^c=eg,f^c=fg54 conjugacy classes — view the full character table.
| maximal subgroup of | #10682705, #10682707, #10682708, … |
| maximal normal subgroup of | #10682705, #10682707, #10682708, … |
| maximal quotient of | #10695786, #10695787, #10695791, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 2589);