Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C3×#431 ⋊ C2 |
| Sylow | C3 ⋊ #2975 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C4 | C23×D3 |
| commutator, socle | C22×C3 | C23×C4 |
| Fitting | C3×#431 | C2 |
| prime radical | C22×D12 | C4 |
| cosocle | C2×C3×C4 | C24 |
| hypercenter | #431 | D3 |
| hypocenter | C3 | #2975 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C2×C4 ◁ #431 | |
452 conjugacy classes of subgroups; only the 177 normal subgroups are shown.
This group has 452 conjugacy classes of subgroups — too many to draw. Only its 177 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) = 96 |
A minimal faithful permutation representation — 4 generators on 19 points:
(1,2,3)(5,9)(8,10);(2,3)(6,11)(8,10)(13,16)(17,19);(2,3)(4,6)(5,8)(7,11)(9,10)(13,16)(17,19);(4,5)(6,10)(7,9)(8,11)(12,13,15,17,14,16,18,19)Polycyclic presentation:
2^7,3|a^2=f,f^2=g,b^a=be,c^a=cg,d^a=deg,c^b=ceg,h^b=h^2,h^c=h^284 conjugacy classes — view the full character table.
| maximal subgroup of | #10689057, #10689060, #10689849, … |
| maximal normal subgroup of | #10689057, #10689060, #10689849, … |
| maximal quotient of | #10793862, #10793934, #10794217, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 11864);