Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C3×#435 ⋊ C2 |
| Sylow | C3 ⋊ #2929 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C4 | C23×D3 |
| commutator, socle | C22×C3 | C23×C4 |
| Fitting | C3×#435 | C2 |
| prime radical | #5547 | C2 |
| cosocle | C2×C3×C4 | C24 |
| hypercenter | #435 | D3 |
| hypocenter | C3 | #2929 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C2×C4 ◁ #435 | |
520 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2)(9,14)(13,19)(15,21)(20,23);(2,3)(8,10,11,16,12,17,18,22)(9,13,15,20,14,19,21,23);(5,7);(4,5)(6,7)(8,9,11,15,12,14,18,21)(10,13,16,20,17,19,22,23)Polycyclic presentation:
2^7,3|a^2=f,c^2=f,f^2=g,b^a=be,d^a=dg,h^c=h^2,h^d=h^284 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#2929, C2×#79478, #11659103, … |
| maximal normal subgroup of | D3×#2929, C2×#79478, #11659103, … |
| maximal quotient of | #10765052, #10765061, #10765070, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 12054);