Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C4×#41 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C22 | C22×S4 |
| commutator, cosocle | C2×SL(2,𝔽3) | C23 |
| Fitting | C4×#41 | D3 |
| hypercenter | C42 | S4 |
| hypocenter | SL(2,𝔽3) | C2×D4 |
| LCS | G ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C22 ◁ C42 | |
329 conjugacy classes of subgroups; only the 42 normal subgroups are shown.
This group has 329 conjugacy classes of subgroups — too many to draw. Only its 42 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 20 points:
(5,17,14,10,15,20,9,11,16,6,7,19)(8,12,13,18);(1,2,3,4);(2,4)(6,10)(7,13)(8,15)(11,12)(14,16)(17,18)Polycyclic presentation:
2^4,3,2^3|b^2=d,c^2=h,f^2=h,g^2=h,b^a=bd,c^a=ch,e^a=e^2,f^a=gh,g^a=fh,f^e=g,g^e=fgh,g^f=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #11711173, #11711207, #11711221, … |
| maximal normal subgroup of | #11711173, #11711207, #11711221, … |
| maximal quotient of | #11712195, #11712196, #11712197, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 17922);