Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Sylow | C3 ⋊ #2365 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator | C22×C3 | C23×C4 |
| Fitting | C3×C4×C4⋊C4 | C2 |
| socle, prime radical, cosocle | C23×C3 | C24 |
| hypercenter | C4×C4⋊C4 | D3 |
| hypocenter | C3 | #2365 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊C4 | |
552 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 19 points:
(1,2)(12,14,15,18)(13,17,16,19);(8,9,10,11);(4,6,7,5)(13,16)(14,18);(2,3)(4,5,7,6)(9,11)(12,13)(14,17)(15,16)(18,19)Polycyclic presentation:
2^7,3|a^2=g,b^2=g,c^2=f,d^2=e,b^a=be,c^a=cf,h^a=h^2,d^b=de,h^d=h^290 conjugacy classes — view the full character table.
| maximal subgroup of | #10816987, #10816989, #10817009, … |
| maximal normal subgroup of | #10816987, #10816989, #10817009, … |
| maximal quotient of | C4⋊C4×Dic3⋊C4, #10716727, #10716760, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 6228);