Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| Sylow | C3 ⋊ #2339 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator | C22×C3 | C23×C4 |
| Fitting, prime radical | C22×C3×#41 | C2 |
| socle, cosocle | C23×C3 | C24 |
| hypercenter | C22×#41 | D3 |
| hypocenter | C3 | #2339 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C22×#41 | |
928 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 19 points:
(1,2,3)(13,16)(17,19);(9,11)(12,14,15,18)(13,17,16,19);(12,13)(14,17)(15,16)(18,19);(2,3)(4,5,6,7)(8,9)(10,11)(13,16)(17,19)Polycyclic presentation:
2^7,3|a^2=g,c^2=e,b^a=be,c^a=cf,h^a=h^2,d^b=de90 conjugacy classes — view the full character table.
| maximal subgroup of | #10692346, #10692350, #10692374, … |
| maximal normal subgroup of | #10692346, #10692350, #10692374, … |
| maximal quotient of | #10716596, #10716612, D4×C3⋊#95, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 16664);