Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C3×#392 ⋊ C2 |
| Sylow | C3 ⋊ #2456 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, cosocle | C23×C3 | C24 |
| Fitting | C3×#392 | C2 |
| hypercenter | #392 | D3 |
| hypocenter | C3 | #2456 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #392 | |
718 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)(5,7)(12,14,15,18)(13,17,16,19);(9,11)(13,16)(17,19);(5,7)(8,9)(10,11);(2,3)(4,5,6,7)(12,13)(14,17)(15,16)(18,19)Polycyclic presentation:
2^7,3|a^2=e,d^2=f,b^a=be,c^a=cf,d^a=de,h^a=h^2,c^b=cg,h^d=h^260 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#2456, C2×#76676, #10962718, … |
| maximal normal subgroup of | D3×#2456, C2×#76676, #10962718, … |
| maximal quotient of | #10717225, #10717271, #10718151, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 9252);