Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| Fitting | C2×C3×C22⋊Q8 ⋊ C2 |
| Sylow | C3 ⋊ #2620 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, cosocle | C23×C3 | C24 |
| Fitting | C2×C3×C22⋊Q8 | C2 |
| prime radical | C2×#930 | C2 |
| hypercenter | C2×C22⋊Q8 | D3 |
| hypocenter | C3 | #2620 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C2×C22⋊Q8 | |
612 conjugacy classes of subgroups; only the 157 normal subgroups are shown.
This group has 612 conjugacy classes of subgroups — too many to draw. Only its 157 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 19 points:
(1,2,3)(12,14,15,18)(13,17,16,19);(2,3)(5,7)(9,11)(14,18)(17,19);(8,9)(10,11)(13,16)(17,19);(2,3)(4,5,6,7)(12,13,15,16)(14,17,18,19)Polycyclic presentation:
2^7,3|a^2=e,d^2=ef,b^a=bef,c^a=cf,h^a=h^2,c^b=cg,d^c=def,h^c=h^260 conjugacy classes — view the full character table.
| maximal subgroup of | #10833422, #10833423, #10833450, … |
| maximal normal subgroup of | #10833422, #10833423, #10833450, … |
| maximal quotient of | #10721560, #10721566, #10721587, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 15397);