Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C3 ⋊ #2764 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, cosocle | C23×C3 | C24 |
| Fitting | C3×#397 | C2 |
| prime radical | C24×C3 | C23 |
| hypercenter | #397 | D3 |
| hypocenter | C3 | #2764 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #397 | |
400 conjugacy classes of subgroups; only the 121 normal subgroups are shown.
This group has 400 conjugacy classes of subgroups — too many to draw. Only its 121 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2)(13,17)(16,18)(20,22,23,27)(21,25,24,26);(2,3)(6,10)(8,11)(12,14,15,19)(13,16,17,18)(20,21)(22,25)(23,24)(26,27);(2,3)(4,6)(5,8)(7,10)(9,11)(20,21,23,24)(22,26,27,25);(2,3)(4,5,7,9)(6,8,10,11)(12,13,15,17)(14,18,19,16)(21,24)(25,26)Polycyclic presentation:
2^7,3|a^2=f,b^2=e,c^2=efg,d^2=e,b^a=be,c^a=cfg,d^a=defg,h^a=h^2,c^b=cg,d^b=de,h^b=h^2,h^c=h^2,h^d=h^254 conjugacy classes — view the full character table.
| maximal subgroup of | #10707839, #10707843, #10708323, … |
| maximal normal subgroup of | #10707839, #10707843, #10708323, … |
| maximal quotient of | #10722803, #10722843, #10722924, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 10189);