Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| Fitting | C3×B(4,𝔽2) ⋊ C2 |
| Sylow | C3 ⋊ #3025 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D3×#120 |
| commutator, cosocle | C23×C3 | C24 |
| Frattini | C23 | C23×D3 |
| Fitting | C3×B(4,𝔽2) | C2 |
| socle | C2×C3 | C2×#120 |
| hypercenter | B(4,𝔽2) | D3 |
| hypocenter | C3 | #3025 |
| LCS | G ▷ C23×C3 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ B(4,𝔽2) | |
766 conjugacy classes of subgroups; only the 133 normal subgroups are shown.
This group has 766 conjugacy classes of subgroups — too many to draw. Only its 133 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) = 48 |
A minimal faithful permutation representation — 4 generators on 19 points:
(2,3)(4,7,8,14)(5,10,11,17)(6,12,13,18)(9,16,15,19);(1,2,3)(4,6)(5,9)(7,12)(8,13)(10,16)(11,15)(14,18)(17,19);(4,5)(6,9)(7,10)(8,11)(12,16)(13,15)(14,17)(18,19);(9,15)(16,19)Polycyclic presentation:
2^7,3|d^2=g,b^a=be,c^a=cf,f^b=fg,e^c=eg,h^d=h^248 conjugacy classes — view the full character table.
| maximal subgroup of | #10949205, #10949207, #10949233, … |
| maximal normal subgroup of | #10949205, #10949207, #10949233, … |
| maximal quotient of | #10767452, #10767454, #10767461, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 12880);