Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | Aut(G) = (#1020⋊C3)⋊C2 (order 576) · Inn(G) = C22⋊A4 · Out(G) = C2×D3 |
| homology | Abelianization (H1) = C3 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting, derived, Sylow | #2811 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, hypercenter | C23 | C22⋊A4 |
| commutator, Fitting, cosocle, hypocenter | #2811 | C3 |
| LCS | G ▷ #2811 | |
| UCS | C1 ◁ C23 | |
142 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,5,3)(6,7,8)(10,13,11)(14,15,16)(18,19,23)(21,22,24);(11,16)(13,15)(17,21,20,23)(18,24,22,19);(1,7,4,5)(2,3,6,8)(9,15,12,13)(10,11,14,16)Polycyclic presentation:
3,2^7|b^2=f,c^2=fgh,d^2=fgh,e^2=f,b^a=be,c^a=d,d^a=cdfgh,e^a=b,c^b=cfg,d^b=dfg,e^b=ef,d^c=dfgh,e^c=efg32 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5867);