Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C3 ⋊ #2845 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, prime radical, cosocle | C23×C3 | C24 |
| Fitting | C3×#382 | C2 |
| hypercenter | #382 | D3 |
| hypocenter | C3 | #2845 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #382 | |
340 conjugacy classes of subgroups; only the 133 normal subgroups are shown.
This group has 340 conjugacy classes of subgroups — too many to draw. Only its 133 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:
(1,2,3)(6,10)(8,11)(12,15,16,14)(13,17,18,19);(5,9)(8,11)(13,18)(14,15)(20,22,23,27)(21,24,25,26);(2,3)(4,6)(5,8)(7,10)(9,11)(12,14,16,15)(13,17,18,19)(20,21,23,25)(22,26,27,24);(4,5,7,9)(6,8,10,11)(12,13)(14,19)(15,17)(16,18)Polycyclic presentation:
2^7,3|a^2=e,b^2=g,c^2=efg,d^2=ef,b^a=bef,c^a=cf,c^b=cg,d^b=de,h^b=h^2,d^c=def54 conjugacy classes — view the full character table.
| maximal subgroup of | #10698265, #10698266, #10698268, … |
| maximal normal subgroup of | #10698265, #10698266, #10698268, … |
| maximal quotient of | #10726010, #10726012, #10726014, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 6795);