Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| Fitting | C2×C3×2−1+4 ⋊ C2 |
| Sylow | C3 ⋊ #3586 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C23×#67 |
| commutator, cosocle | C3×C4 | C25 |
| Frattini | C4 | C24×D3 |
| Fitting | C2×C3×2−1+4 | C2 |
| socle | C2×C3 | C23×D4 |
| hypercenter | C2×2−1+4 | D3 |
| hypocenter | C3 | #3586 |
| LCS | G ▷ C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2 ◁ C2×#41 ◁ C2×2−1+4 | |
1412 conjugacy classes of subgroups; only the 545 normal subgroups are shown.
This group has 1412 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)(5,12)(10,24)(11,25)(13,20)(14,27)(23,34)(26,32)(28,35);(7,18)(11,25)(15,29)(19,31)(23,34)(26,32)(28,35)(30,33);(4,7)(5,11)(6,15)(8,19)(9,18)(10,23)(12,25)(13,26)(14,28)(16,30)(17,29)(20,32)(21,33)(22,31)(24,34)(27,35);(4,6,9,17)(5,10,12,24)(7,15,18,29)(8,21,22,16)(11,23,25,34)(13,27,20,14)(19,33,31,30)(26,35,32,28);(2,3)(4,5)(6,14)(7,11)(8,20)(9,12)(10,16)(13,22)(15,28)(17,27)(18,25)(19,32)(21,24)(23,30)(26,31)(29,35)(33,34)Polycyclic presentation:
2^7,3|b^2=g,f^2=g,b^a=bf,e^a=eg,f^a=fg,h^a=h^2,f^b=fg,d^c=dg75 conjugacy classes — view the full character table.
| maximal subgroup of | #10826492, #10826493, #10826514, … |
| maximal normal subgroup of | #10826492, #10826493, #10826514, … |
| maximal quotient of | #11658764, #11658770, #11658871, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 19863);