Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C3 ⋊ #2586 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, prime radical, cosocle | C23×C3 | C24 |
| Fitting | C3×#395 | C2 |
| hypercenter | #395 | D3 |
| hypocenter | C3 | #2586 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #395 | |
412 conjugacy classes of subgroups; only the 149 normal subgroups are shown.
This group has 412 conjugacy classes of subgroups — too many to draw. Only its 149 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) = 384 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2)(5,9)(8,10)(12,14,15,18)(13,17,16,19)(20,23,24,22)(21,26,27,25);(2,3)(6,11)(8,10)(13,16)(17,19)(20,22,24,23)(21,25,27,26);(4,6)(5,8)(7,11)(9,10)(20,22,24,23)(21,25,27,26);(2,3)(4,5,7,9)(6,10,11,8)(12,13)(14,17)(15,16)(18,19)(20,21,24,27)(22,26,23,25)Polycyclic presentation:
2^7,3|a^2=e,b^2=eg,c^2=eg,d^2=f,b^a=be,c^a=cf,d^a=de,h^a=h^2,c^b=cg,h^c=h^2,h^d=h^260 conjugacy classes — view the full character table.
| maximal subgroup of | #10707869, #10707873, #10708353, … |
| maximal normal subgroup of | #10707869, #10707873, #10708353, … |
| maximal quotient of | #10717639, #10719056, #10719086, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 9770);