Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C3 ⋊ #2855 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, cosocle | C23×C3 | C24 |
| Fitting | C3×#385 | C2 |
| prime radical | C24×C3 | C23 |
| hypercenter | #385 | D3 |
| hypocenter | C3 | #2855 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #385 | |
564 conjugacy classes of subgroups; only the 165 normal subgroups are shown.
This group has 564 conjugacy classes of subgroups — too many to draw. Only its 165 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2,3)(5,7)(16,18,19,23)(17,21,20,22);(9,13)(12,15)(17,20)(21,22);(2,3)(8,10,11,14)(9,12,13,15)(17,20)(21,22);(4,5)(6,7)(8,9)(10,12)(11,13)(14,15)(16,17,19,20)(18,22,23,21)Polycyclic presentation:
2^7,3|a^2=e,b^2=ef,d^2=e,b^a=be,c^a=cf,d^a=dg,h^b=h^272 conjugacy classes — view the full character table.
| maximal subgroup of | #10829382, #10829386, #10831858, … |
| maximal normal subgroup of | #10829382, #10829386, #10831858, … |
| maximal quotient of | #10726368, #10726406, #10726412, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 7165);