Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | #543 ⋊ D3 |
| derived | #543⋊C3 ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, hypercenter | C22 | #1020 |
| commutator, cosocle, hypocenter | #543⋊C3 | C2 |
| Fitting | #543 | D3 |
| LCS | G ▷ #543⋊C3 | |
| UCS | C1 ◁ C22 | |
170 conjugacy classes of subgroups; only the 12 normal subgroups are shown.
This group has 170 conjugacy classes of subgroups — too many to draw. Only its 12 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) = 16 |
A minimal faithful permutation representation — 2 generators on 16 points:
(1,15,7,3,5,8,11,6)(2,16,13,4,9,14,10,12);(2,7,3)(6,9,11)(8,14,10)(13,15,16)Polycyclic presentation:
2,3,2^6|e^2=h,f^2=h,b^a=b^2,c^a=d,d^a=c,e^a=fh,f^a=eh,c^b=d,d^b=cd,e^b=ef,f^b=e,e^c=eg,f^d=fg,f^e=fh21 conjugacy classes — view the full character table.
| maximal subgroup of | #11711401, #11711403, #11711405, … |
| maximal normal subgroup of | #11711401, #11711403, #11711405, … |
| maximal quotient of | #11714220, #11714221, #11714229, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18127);