Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| unambiguous semidirect product | #3497⋊C3 |
| Fitting, Sylow | #3497 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C22 | C23×A4 |
| commutator, cosocle | C2×Q8 | C23×C3 |
| Fitting | #3497 | C3 |
| hypercenter | #121 | A4 |
| hypocenter | Q8 | C2×C3×D4 |
| LCS | G ▷ C2×Q8 ▷ Q8 | |
| UCS | C1 ◁ C22 ◁ #121 | |
372 conjugacy classes of subgroups; only the 62 normal subgroups are shown.
This group has 372 conjugacy classes of subgroups — too many to draw. Only its 62 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:
(6,10)(7,13,8)(11,20,12,17,19,18)(14,15,16);(1,2,3,4)(5,15,9,13)(6,20,10,19)(7,8,14,16)(11,12,17,18);(2,4)(5,6,9,10)(7,11,14,17)(8,12,16,18)(13,19,15,20)Polycyclic presentation:
2^3,3,2^4|a^2=h,b^2=e,f^2=h,g^2=h,b^a=be,c^a=ch,f^d=fg,g^d=f,g^f=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #17618⋊C3, #61817⋊C3, #61820⋊C3, … |
| maximal normal subgroup of | #17618⋊C3, #61817⋊C3, #61820⋊C3, … |
| maximal quotient of | #34182⋊C3, #34257⋊C3, #34311⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18185);