Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | #585 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | #1020 |
| commutator, cosocle, hypocenter | #997 | C22 |
| Frattini, socle | C2 | C2×#1020 |
| Fitting | #585 | D3 |
| LCS | G ▷ #997 | |
| UCS | C1 ◁ C4 | |
267 conjugacy classes of subgroups; only the 16 normal subgroups are shown.
This group has 267 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:
(1,13,11,8,9,7,5,16,4,2,12,15)(3,6,10,14);(3,12,10,9)(6,16,14,13)Polycyclic presentation:
2^2,3,2^5|b^2=h,f^2=h,g^2=h,c^a=c^2,e^a=de,f^a=fg,g^a=gh,d^c=e,e^c=de,f^c=g,g^c=fgh,f^d=fh,f^e=fh,g^e=gh,g^f=gh26 conjugacy classes — view the full character table.
| maximal subgroup of | #11711596, #11711597, #11714162, … |
| maximal normal subgroup of | #11711596, #11711597, #11714162, … |
| maximal quotient of | #11715575, #11715578, #11715579, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 20096);