Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| Fitting | C3×D4×Q8 ⋊ C2 |
| Sylow | C3 ⋊ #3523 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D3 |
| commutator, socle, cosocle | C22×C3 | C25 |
| Fitting | C3×D4×Q8 | C2 |
| prime radical | #939 | C22 |
| hypercenter | D4×Q8 | D3 |
| hypocenter | C3 | #3523 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ D4×Q8 | |
992 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 5 generators on 27 points:
(2,3)(4,6,7,10)(5,9,8,11)(13,18)(17,20)(19,26)(25,27);(1,2,3)(12,15,16,24)(13,19,18,26)(14,21,22,23)(17,25,20,27);(5,8)(9,11)(13,18)(17,20)(19,26)(25,27);(12,14,16,22)(13,17,18,20)(15,23,24,21)(19,27,26,25);(4,5)(6,9)(7,8)(10,11)(12,13,16,18)(14,20,22,17)(15,19,24,26)(21,27,23,25)Polycyclic presentation:
2^7,3|a^2=f,b^2=f,d^2=f,e^2=fg,b^a=bf,c^a=cg,e^a=ef,d^b=df,h^e=h^269 conjugacy classes — view the full character table.
| maximal subgroup of | #11677763, #11677767, #11677826, … |
| maximal normal subgroup of | #11677763, #11677767, #11677826, … |
| maximal quotient of | #10955419, #10955926, #10956042, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 19328);