Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| Fitting | C2×C3×#131 ⋊ C2 |
| Sylow | C3 ⋊ #3584 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | C22×D12 |
| commutator, cosocle | C3×C4 | C25 |
| Frattini | C4 | C24×D3 |
| Fitting | C2×C3×#131 | C2 |
| socle | C2×C3 | C23×D4 |
| hypercenter | C2×#131 | D3 |
| hypocenter | C3 | #3584 |
| LCS | G ▷ C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C4 ◁ C2×#41 ◁ C2×#131 | |
1424 conjugacy classes of subgroups; only the 545 normal subgroups are shown.
This group has 1424 conjugacy classes of subgroups — too many to draw. Only its 545 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 — 5 generators on 35 points:
(1,2,3)(4,8,10,9)(5,13,14,22)(6,17,18,23)(7,20,19,21)(11,25,26,15)(12,28,27,32)(16,31,30,33)(24,35,34,29);(7,19)(12,27)(16,30)(20,21)(24,34)(28,32)(29,35)(31,33);(4,7)(5,12)(6,16)(8,20)(9,21)(10,19)(11,24)(13,28)(14,27)(15,29)(17,31)(18,30)(22,32)(23,33)(25,35)(26,34);(2,3)(4,6)(5,11)(7,16)(8,17)(9,23)(10,18)(12,24)(13,25)(14,26)(15,22)(19,30)(20,31)(21,33)(27,34)(28,35)(29,32);(2,3)(4,5)(6,15)(7,12)(8,13)(9,22)(10,14)(11,17)(16,29)(18,25)(19,27)(20,28)(21,32)(23,26)(24,31)(30,35)(33,34)Polycyclic presentation:
2^7,3|e^2=g,f^2=g,b^a=bf,f^a=fg,h^a=h^2,f^b=fg,h^b=h^2,d^c=dg84 conjugacy classes — view the full character table.
| maximal subgroup of | #10710892, #10710895, #10710897, … |
| maximal normal subgroup of | #10710892, #10710895, #10710897, … |
| maximal quotient of | #11658753, #11658866, #11659308, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 19739);