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 ⋊ #1445 |
| 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 | #1445 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C24 ◁ C22×C4⋊C4 | |
520 conjugacy classes of subgroups; only the 197 normal subgroups are shown.
This group has 520 conjugacy classes of subgroups — too many to draw. Only its 197 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2,3)(9,11)(13,15)(16,19,20,18)(17,21,22,23);(9,11)(12,13)(14,15)(16,18,20,19)(17,21,22,23);(2,3)(4,5,6,7)(8,9)(10,11)(16,17)(18,23)(19,21)(20,22)Polycyclic presentation:
2^7,3|a^2=g,b^2=de,c^2=de,b^a=bd,c^a=ce,h^a=h^2,c^b=cf84 conjugacy classes — view the full character table.
| maximal subgroup of | #10682509, #10687319, #10687321, … |
| maximal normal subgroup of | #10682509, #10687319, #10687321, … |
| maximal quotient of | #10683111, #10683114, #10703917, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 4145);