Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C42 · Schur multiplier (H2) = C4 |
No aliases for this group.
| Fitting, Sylow | #2276 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C23 | C22×A4 |
| commutator, hypocenter | Q8 | C3×C42 |
| Fitting | #2276 | C3 |
| prime radical | C22×SL(2,𝔽3) | C22 |
| cosocle | C22×Q8 | C22×C3 |
| hypercenter | #95 | A4 |
| LCS | G ▷ Q8 | |
| UCS | C1 ◁ C23 ◁ #95 | |
193 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 24 points:
(5,6,7,8)(10,14)(12,19,13)(15,24,16,21,23,22)(17,20,18);(1,2,3,4)(9,23,11,24)(10,19,14,20)(12,16,17,22)(13,21,18,15)Polycyclic presentation:
2^2,3,2^5|a^2=e,b^2=dh,f^2=h,g^2=h,b^a=bh,f^c=fgh,g^c=f,g^f=gh76 conjugacy classes — view the full character table.
| maximal subgroup of | #17549⋊C3, #17550⋊C3, C2×#73167, … |
| maximal normal subgroup of | #17549⋊C3, #17550⋊C3, C2×#73167, … |
| maximal quotient of | SL(2,𝔽3)×#95, #20569⋊C3, #20572⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5743);