Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C8 |
No aliases for this group.
| Sylow | C3 ⋊ #1453 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C42 | C22×D3 |
| commutator, socle | C22×C3 | C2×C42 |
| Fitting | C3×#431 | C2 |
| prime radical | C23×C3 | C42 |
| cosocle | C3×C42 | C23 |
| hypercenter | #431 | D3 |
| hypocenter | C3 | #1453 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C42 ◁ #431 | |
190 conjugacy classes of subgroups; only the 105 normal subgroups are shown.
This group has 190 conjugacy classes of subgroups — too many to draw. Only its 105 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 — 3 generators on 27 points:
(1,2,3)(5,10)(6,11)(14,20)(17,23)(21,26)(24,27);(4,6,8,10,9,11,7,5)(12,14)(13,17)(15,21)(16,20)(18,24)(19,23)(22,26)(25,27);(2,3)(4,5,7,11,9,10,8,6)(12,13,15,18,16,19,22,25)(14,17,21,24,20,23,26,27)Polycyclic presentation:
2^7,3|a^2=d,b^2=e,d^2=f,e^2=g,c^a=cg,h^a=h^2,c^b=cf96 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#1453, C2×#70743, #10760819, … |
| maximal normal subgroup of | D3×#1453, C2×#70743, #10760819, … |
| maximal quotient of | #10684555, #10684575, #10685619, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 3319);