Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C3×#501 ⋊ C2 |
| Sylow | C3 ⋊ #3179 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×D3×D4 |
| commutator, cosocle | C2×C3×C4 | C24 |
| Frattini | C2×C4 | C23×D3 |
| Fitting | C3×#501 | C2 |
| socle | C22×C3 | C22×D4 |
| hypercenter | #501 | D3 |
| hypocenter | C3 | #3179 |
| LCS | G ▷ C2×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #501 | |
374 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2)(4,6,7,10)(5,9,8,11)(12,15,16,13)(14,22,21,18)(17,24,25,19)(20,23,27,26);(12,14,17,23,16,21,25,26)(13,18,19,20,15,22,24,27);(2,3)(5,8)(9,11)(13,15)(14,21)(19,24)(23,26);(2,3)(4,5)(6,9)(7,8)(10,11)(12,13,16,15)(14,20,21,27)(17,24,25,19)(18,26,22,23)Polycyclic presentation:
2^7,3|a^2=g,c^2=f,d^2=e,f^2=g,b^a=be,c^a=cf,f^a=fg,h^a=h^2,c^b=cg,d^b=dg,h^b=h^2,h^d=h^248 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#3179, C2×#81961, #11665079, … |
| maximal normal subgroup of | D3×#3179, C2×#81961, #11665079, … |
| maximal quotient of | #10795765, #10799478, #10799486, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 14537);