Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #3358⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | #3358⋊C3 |
| Fitting, Sylow | #3358 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D4×A4 |
| commutator, cosocle | C4×Q8 | C22×C3 |
| Frattini | C2×C4 | C22×A4 |
| Fitting | #3358 | C3 |
| hypercenter | #102 | A4 |
| hypocenter | Q8 | C3×D8 |
| LCS | G ▷ C4×Q8 ▷ C2×Q8 ▷ Q8 | |
| UCS | C1 ◁ C22 ◁ C2×C4 ◁ #102 | |
131 conjugacy classes of subgroups; only the 28 normal subgroups are shown.
This group has 131 conjugacy classes of subgroups — too many to draw. Only its 28 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 — 2 generators on 24 points:
(2,5)(3,8)(6,7)(11,17,12)(14,21,15)(18,19,20)(22,23,24);(1,2)(3,5)(4,7)(6,8)(9,23)(10,17)(11,15)(12,22)(13,21)(14,20)(16,19)(18,24)Polycyclic presentation:
2^2,3,2^5|a^2=h,d^2=e,f^2=h,g^2=h,b^a=bd,d^a=de,d^b=de,f^c=fg,g^c=f,g^f=gh49 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#3358⋊C3, #61303⋊C3, #61317⋊C3, … |
| maximal normal subgroup of | C2×#3358⋊C3, #61303⋊C3, #61317⋊C3, … |
| maximal quotient of | SL(2,𝔽3)×#102, #22540⋊C3, #22740⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5839);