Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #4656 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Sylow | C3 ⋊ #1441 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C24 | C22×D3 |
| commutator | C23×C3 | C22×C4 |
| Fitting | C22×C3×C4⋊C4 | C2 |
| socle, prime radical, cosocle | C24×C3 | C23 |
| hypercenter | C22×C4⋊C4 | D3 |
| hypocenter | C3 | #1441 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C24 ◁ C22×C4⋊C4 | |
564 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 3 generators on 19 points:
(2,3)(9,11)(13,15)(16,18,19,17);(1,2,3)(12,13)(14,15)(16,17,19,18);(4,5,6,7)(8,9)(10,11)(17,18)Polycyclic presentation:
2^7,3|a^2=g,b^2=d,c^2=d,b^a=bd,c^a=ce,c^b=cf,h^c=h^284 conjugacy classes — view the full character table.
| maximal subgroup of | #10686994, #10687004, #10687009, … |
| maximal normal subgroup of | #10686994, #10687004, #10687009, … |
| maximal quotient of | #10683110, #10683113, #10683114, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 4142);