Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| Fitting | #585 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C2 | C2×#1020 |
| commutator, cosocle, hypocenter | #997 | C22 |
| Fitting | #585 | D3 |
| hypercenter | C4 | #1020 |
| LCS | G ▷ #997 | |
| UCS | C1 ◁ C2 ◁ C4 | |
296 conjugacy classes of subgroups; only the 16 normal subgroups are shown.
This group has 296 conjugacy classes of subgroups — too many to draw. Only its 16 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:
(1,2,5,6)(3,8,9,12,4,15,10,13,16,7,11,14);(1,9,3,4)(2,15,7,13)(5,16,10,11)(6,14,12,8)Polycyclic presentation:
2^2,3,2^5|b^2=h,b^a=bh,c^a=c^2,e^a=de,f^a=fg,d^c=e,e^c=de,f^c=g,g^c=fg,f^d=fh,f^e=fh,g^e=gh23 conjugacy classes — view the full character table.
| maximal subgroup of | #11711638, #11711639, #11711643, … |
| maximal normal subgroup of | #11711638, #11711639, #11711643, … |
| maximal quotient of | #11715597, #11715598, #11715599, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 20097);