Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×S4 |
| commutator, cosocle | C22×A4 | C23 |
| Fitting | C23×D4 | D3 |
| socle | C24 | C22×D3 |
| prime radical | C2×D4×A4 | C2 |
| hypercenter | C2×D4 | S4 |
| hypocenter | A4 | #124 |
| LCS | G ▷ C22×A4 ▷ A4 | |
| UCS | C1 ◁ C22 ◁ C2×D4 | |
411 conjugacy classes of subgroups; only the 44 normal subgroups are shown.
This group has 411 conjugacy classes of subgroups — too many to draw. Only its 44 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 3 generators on 16 points:
(2,4)(5,6,8)(10,14)(11,15);(9,11,12,15)(10,13,14,16);(1,2,3,4)(6,7)(9,10)(11,13)(12,14)(15,16)Polycyclic presentation:
2^5,3,2^2|a^2=e,b^2=d,c^a=cde,f^a=f^2,g^a=h,h^a=g,c^b=cd,g^f=h,h^f=gh38 conjugacy classes — view the full character table.
| maximal subgroup of | #11711615, #11713181, #11713319, … |
| maximal normal subgroup of | #11711615, #11713181, #11713319, … |
| maximal quotient of | #11711702, #11711708, #11711712, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18068);