Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #4656 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C3 ⋊ #1917 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22×C4 | C22×D3 |
| commutator, socle, prime radical | C23×C3 | C22×C4 |
| Fitting | C3×#336 | C2 |
| cosocle | C22×C3×C4 | C23 |
| hypercenter | #336 | D3 |
| hypocenter | C3 | #1917 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C22×C4 ◁ #336 | |
228 conjugacy classes of subgroups; only the 97 normal subgroups are shown.
This group has 228 conjugacy classes of subgroups — too many to draw. Only its 97 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)(5,7)(8,10,11,14)(9,13,12,15)(16,19,20,21)(17,18,22,23);(9,12)(13,15)(16,18,21,22,20,23,19,17);(2,3)(4,5)(6,7)(8,9)(10,13)(11,12)(14,15)(16,17,21,18,20,22,19,23)Polycyclic presentation:
2^7,3|a^2=f,b^2=f,c^2=d,f^2=g,b^a=bd,c^a=ce,h^a=h^2,c^b=cg,h^c=h^284 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#1917, C2×#68612, #10797249, … |
| maximal normal subgroup of | D3×#1917, C2×#68612, #10797249, … |
| maximal quotient of | #10693333, #10693340, #10693347, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 1188);