Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #68629 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C3×#336 ⋊ C2 |
| Sylow | C3 ⋊ #2011 |
| subgroup | G/H | |
|---|---|---|
| center | C23 | C2×D12 |
| commutator, cosocle | C22×C3×C4 | C23 |
| Frattini | C22×C4 | C22×D3 |
| Fitting | C3×#336 | C2 |
| socle | C23×C3 | C2×D4 |
| hypercenter | #336 | D3 |
| hypocenter | C3 | #2011 |
| LCS | G ▷ C22×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ #336 | |
510 conjugacy classes of subgroups; only the 99 normal subgroups are shown.
This group has 510 conjugacy classes of subgroups — too many to draw. Only its 99 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,2)(9,11)(13,18)(14,15)(17,19);(2,3)(5,7)(8,9)(10,11)(13,17)(14,15)(18,19);(2,3)(4,5)(6,7)(12,13)(14,17)(15,18)(16,19)Polycyclic presentation:
2^7,3|d^2=g,e^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eg,h^a=h^2,c^b=cf,d^b=dg,e^b=eg,h^b=h^2,d^c=dg,e^c=eg,h^c=h^272 conjugacy classes — view the full character table.
| maximal subgroup of | #10683886, #10683891, #10683893, … |
| maximal normal subgroup of | #10683886, #10683891, #10683893, … |
| maximal quotient of | #10695610, #10695652, #10695720, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 1250);