Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| Sylow | C3 ⋊ #2922 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C4 | C23×D3 |
| commutator, socle | C22×C3 | C23×C4 |
| Fitting | C3×#408 | C2 |
| prime radical | C2×C3×#41 | C22 |
| cosocle | C2×C3×C4 | C24 |
| hypercenter | #408 | D3 |
| hypocenter | C3 | #2922 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C2×C4 ◁ #408 | |
400 conjugacy classes of subgroups; only the 177 normal subgroups are shown.
This group has 400 conjugacy classes of subgroups — too many to draw. Only its 177 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:
(1,2,3)(14,20)(17,23)(21,26)(24,27);(2,3)(4,6,7,10)(5,9,8,11);(5,8)(9,11)(12,14)(13,17)(15,21)(16,20)(18,24)(19,23)(22,26)(25,27);(4,5)(6,9)(7,8)(10,11)(12,13,15,18,16,19,22,25)(14,17,21,24,20,23,26,27)Polycyclic presentation:
2^7,3|a^2=f,c^2=e,f^2=g,b^a=be,d^b=dg,h^c=h^284 conjugacy classes — view the full character table.
| maximal subgroup of | #10704395, #10704399, #10704403, … |
| maximal normal subgroup of | #10704395, #10704399, #10704403, … |
| maximal quotient of | #10792669, Dic3×#408, #10793073, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 11224);