Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| Fitting | C3×#386 ⋊ C2 |
| Sylow | C3 ⋊ #2471 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, cosocle | C23×C3 | C24 |
| Fitting | C3×#386 | C2 |
| prime radical | #5223 | C2 |
| hypercenter | #386 | D3 |
| hypocenter | C3 | #2471 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #386 | |
582 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2,3)(9,13)(12,14)(16,18,19,23)(17,21,20,22);(5,7)(17,20)(21,22);(2,3)(4,5)(6,7)(8,10,11,15)(9,12,13,14)(17,20)(21,22);(2,3)(8,9,11,13)(10,14,15,12)(16,17,19,20)(18,22,23,21)Polycyclic presentation:
2^7,3|a^2=e,b^2=ef,d^2=f,b^a=be,c^a=cf,d^a=de,h^a=h^2,c^b=cg,h^b=h^266 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#2471, C2×#74724, #10963779, … |
| maximal normal subgroup of | D3×#2471, C2×#74724, #10963779, … |
| maximal quotient of | #10717333, #10717763, #10718154, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 7300);