Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×#1020 |
| commutator, prime radical, cosocle | C2×C22⋊A4 | C22 |
| Fitting | C24×C4 | D3 |
| socle | C25 | C2×D3 |
| hypercenter | C4 | #1020 |
| hypocenter | C22⋊A4 | Q8 |
| LCS | G ▷ C2×C22⋊A4 ▷ C22⋊A4 | |
| UCS | C1 ◁ C2 ◁ C4 | |
347 conjugacy classes of subgroups; only the 21 normal subgroups are shown.
This group has 347 conjugacy classes of subgroups — too many to draw. Only its 21 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) = 24 |
A minimal faithful permutation representation — 2 generators on 16 points:
(2,3,4)(5,8,7)(9,11,12,16)(10,13,14,15);(1,4,2,3)(6,8)(9,10,12,14)(11,15,16,13)Polycyclic presentation:
2^3,3,2^4|a^2=c,b^2=c,b^a=bc,d^a=d^2,f^a=ef,g^a=gh,e^d=f,f^d=ef,g^d=h,h^d=gh28 conjugacy classes — view the full character table.
| maximal subgroup of | #11711265, #11711266, #11711267, … |
| maximal normal subgroup of | #11711265, #11711266, #11711267, … |
| maximal quotient of | C4⋊#5936, #11715571, #11715741, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 20078);