Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #548⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #3480⋊C3 |
| Fitting, Sylow | #3480 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×A4 |
| commutator, cosocle | C2×Q8 | C23×C3 |
| Frattini, socle | C22 | C23×A4 |
| Fitting | #3480 | C3 |
| hypercenter | #117 | A4 |
| hypocenter | Q8 | C3×#41 |
| LCS | G ▷ C2×Q8 ▷ Q8 | |
| UCS | C1 ◁ C2×C4 ◁ #117 | |
282 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 282 conjugacy classes of subgroups — too many to draw. Only its 64 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) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,3,4,7)(2,6,5,8)(11,17,20)(12,18,19)(14,21,24)(15,22,23);(2,5)(6,8)(9,20,13,18)(10,24,16,22)(11,12,19,17)(14,15,23,21);(1,2)(3,6)(4,5)(7,8)(9,10,13,16)(11,14,19,23)(12,15,17,21)(18,22,20,24)Polycyclic presentation:
2^3,3,2^4|a^2=h,c^2=e,f^2=h,g^2=h,b^a=be,f^d=gh,g^d=fg,g^f=gh70 conjugacy classes — view the full character table.
| maximal subgroup of | #22550⋊C3, #60218⋊C3, #60274⋊C3, … |
| maximal normal subgroup of | #22550⋊C3, #60218⋊C3, #60274⋊C3, … |
| maximal quotient of | SL(2,𝔽3)×#117, #34125⋊C3, C4×#548⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18178);