Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C22 | C22×S4 |
| commutator, prime radical, cosocle | C2×SL(2,𝔽3) | C23 |
| Fitting | #557 | D3 |
| hypercenter | C4⋊C4 | S4 |
| hypocenter | SL(2,𝔽3) | C2×Q8 |
| LCS | G ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C22 ◁ C4⋊C4 | |
189 conjugacy classes of subgroups; only the 36 normal subgroups are shown.
This group has 189 conjugacy classes of subgroups — too many to draw. Only its 36 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,3,4,8)(2,5,6,7)(9,21,22,10,17,24)(11,15,20,14,12,23)(13,16)(18,19);(1,2,4,6)(3,7,8,5)(9,10,11,14)(12,15,17,21)(13,16,18,19)(20,23,22,24);(9,10,11,14)(12,19,17,16)(13,21,18,15)(20,24,22,23)Polycyclic presentation:
2^4,3,2^3|a^2=h,b^2=d,c^2=dh,f^2=h,g^2=h,c^a=ch,e^a=e^2,f^a=gh,g^a=fh,c^b=cd,f^e=g,g^e=fgh,g^f=gh34 conjugacy classes — view the full character table.
| maximal subgroup of | #11712281, #11712283, #11712320, … |
| maximal normal subgroup of | #11712281, #11712283, #11712320, … |
| maximal quotient of | #11712975, #11713013, #11713018, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 17991);