Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Sylow | C3 ⋊ C4×#123 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator | C22×C3 | C23×C4 |
| Fitting | C3×#386 | C2 |
| socle, cosocle | C23×C3 | C24 |
| prime radical | C22×C3×D4 | C22 |
| hypercenter | #386 | D3 |
| hypocenter | C3 | C4×#123 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #386 | |
688 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)(4,6,7,5);(2,3)(9,11)(12,14,15,18)(13,17,16,19);(13,16)(17,19);(2,3)(4,5,7,6)(8,9)(10,11)(12,13)(14,17)(15,16)(18,19)Polycyclic presentation:
2^7,3|a^2=g,c^2=e,d^2=g,b^a=be,c^a=cf,h^a=h^2,h^c=h^2,h^d=h^284 conjugacy classes — view the full character table.
| maximal subgroup of | #10710323, #10711154, #10801173, … |
| maximal normal subgroup of | #10710323, #10711154, #10801173, … |
| maximal quotient of | Dic3×#386, #10716348, #10716351, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 7237);