Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×S4 |
| commutator, prime radical, cosocle | C22×A4 | C23 |
| Fitting | C22×C42 | D3 |
| socle | C24 | C22×D3 |
| hypercenter | C42 | S4 |
| hypocenter | A4 | C4⋊Q8 |
| LCS | G ▷ C22×A4 ▷ A4 | |
| UCS | C1 ◁ C22 ◁ C42 | |
317 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 317 conjugacy classes of subgroups — too many to draw. Only its 56 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) = 96 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2,3,4)(5,6,8);(9,11,12,16)(10,13,14,15);(2,4)(6,7)(9,10,12,14)(11,15,16,13)Polycyclic presentation:
2^5,3,2^2|a^2=e,b^2=e,c^2=d,b^a=be,c^a=cd,f^a=f^2,g^a=h,h^a=g,g^f=h,h^f=gh44 conjugacy classes — view the full character table.
| maximal subgroup of | #11711156, #11711157, #11711158, … |
| maximal normal subgroup of | #11711156, #11711157, #11711158, … |
| maximal quotient of | #11711739, #11711744, #11711746, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 17859);