Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C3 ⋊ #2030 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | #937 |
| commutator, prime radical, cosocle | C24×C3 | C23 |
| Frattini | C24 | C22×D3 |
| Fitting | C3×#381 | C2 |
| socle | C22×C3 | #120 |
| hypercenter | #381 | D3 |
| hypocenter | C3 | #2030 |
| LCS | G ▷ C24×C3 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ #381 | |
290 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 290 conjugacy classes of subgroups — too many to draw. Only its 69 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) = 96 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,3)(12,15,16,22)(13,18,19,26)(14,20,21,27)(17,23,24,25);(4,6,7,11)(5,8,9,10)(12,14)(13,17)(15,20)(16,21)(18,25)(19,24)(22,27)(23,26);(2,3)(4,5,7,9)(6,10,11,8)(12,13)(14,17)(15,18)(16,19)(20,25)(21,24)(22,26)(23,27)Polycyclic presentation:
2^7,3|a^2=d,b^2=d,c^2=g,b^a=bd,c^a=ce,f^a=fg,h^a=h^2,c^b=cf,e^b=eg54 conjugacy classes — view the full character table.
| maximal subgroup of | #10682768, #10682770, #10707915, … |
| maximal normal subgroup of | #10682768, #10682770, #10707915, … |
| maximal quotient of | #10698165, #10698167, #10698170, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 2112);