Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Fitting | C3×H(3,𝔽4) ⋊ C2 |
| Sylow | C3 ⋊ #1655 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | #953 |
| commutator, cosocle | C3×C42 | C23 |
| Frattini | C42 | C22×D3 |
| Fitting | C3×H(3,𝔽4) | C2 |
| socle | C22×C3 | #120 |
| hypercenter | H(3,𝔽4) | D3 |
| hypocenter | C3 | #1655 |
| LCS | G ▷ C3×C42 ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ H(3,𝔽4) | |
262 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,2,3)(6,10)(8,11)(12,13)(14,19)(15,17)(16,18);(4,5)(6,11)(7,9)(8,10)(14,18)(16,19);(2,3)(5,8)(6,10)(9,11)(13,16)(14,18)(17,19)Polycyclic presentation:
2^7,3|d^2=f,e^2=fg,b^a=bd,c^a=ce,d^a=df,e^a=efg,h^a=h^2,d^b=df,e^b=eg,d^c=dg,e^c=efg39 conjugacy classes — view the full character table.
| maximal subgroup of | #10715916, #10715917, D3×#1655, … |
| maximal normal subgroup of | #10715916, #10715917, D3×#1655, … |
| maximal quotient of | #10683885, #10683932, #10683945, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 4524);