Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| Sylow | C3 ⋊ #1831 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22×C4 | C22×D3 |
| commutator | C22×C3 | C22×C8 |
| Fitting | C3×C43 | C2 |
| socle, prime radical | C23×C3 | C22×C4 |
| cosocle | C22×C3×C4 | C23 |
| hypercenter | C43 | D3 |
| hypocenter | C3 | #1831 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22×C4 ◁ C43 | |
260 conjugacy classes of subgroups; only the 149 normal subgroups are shown.
This group has 260 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) = 23 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2,3)(4,6,7,5)(17,21)(20,23);(4,5,7,6)(16,18,19,22)(17,20,21,23);(2,3)(5,6)(8,9,10,12,11,13,14,15)(16,17)(18,20)(19,21)(22,23)Polycyclic presentation:
2^7,3|a^2=f,b^2=e,c^2=d,f^2=g,b^a=bd,c^a=ce,h^a=h^2120 conjugacy classes — view the full character table.
| maximal subgroup of | #10681720, #10681721, #10681730, … |
| maximal normal subgroup of | #10681720, #10681721, #10681730, … |
| maximal quotient of | #10688833, #10688837, #10688848, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 1968);