Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| unambiguous semidirect product | #3486⋊C3 |
| Fitting, Sylow | #3486 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C22 | C23×A4 |
| commutator, cosocle | C2×Q8 | C23×C3 |
| Fitting | #3486 | C3 |
| hypercenter | #120 | A4 |
| hypocenter | Q8 | C2×C3×D4 |
| LCS | G ▷ C2×Q8 ▷ Q8 | |
| UCS | C1 ◁ C22 ◁ #120 | |
402 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 402 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) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 20 points:
(6,10)(7,13,8)(11,20,12,17,19,18)(14,15,16);(2,4)(5,15,9,13)(6,20,10,19)(7,8,14,16)(11,12,17,18);(1,2)(3,4)(5,6)(7,11)(8,12)(9,10)(13,19)(14,17)(15,20)(16,18)Polycyclic presentation:
2^3,3,2^4|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 | #33349⋊C3, #11710349, C2×#3486⋊C3, … |
| maximal normal subgroup of | #33349⋊C3, #11710349, C2×#3486⋊C3, … |
| maximal quotient of | #34173⋊C3, #34249⋊C3, #34257⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18181);