Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C3×#443 ⋊ C2 |
| Sylow | C3 ⋊ #3291 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D4×D12 |
| commutator, cosocle | C2×C3×C4 | C24 |
| Frattini | C2×C4 | C23×D3 |
| Fitting | C3×#443 | C2 |
| socle | C2×C3 | D42 |
| hypercenter | #443 | D3 |
| hypocenter | C3 | #3291 |
| LCS | G ▷ C2×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #443 | |
460 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 460 conjugacy classes of subgroups — too many to draw. Only its 125 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:
(1,2,3)(4,7)(5,12)(6,13)(8,16)(9,18)(10,11)(14,17)(15,19);(2,3)(4,7)(5,6)(8,18)(9,16)(10,11)(12,13)(14,19)(15,17);(4,6)(5,11)(7,15)(8,19)(9,12)(10,17)(13,18)(14,16);(2,3)(4,5,10,15)(6,16,17,18)(7,12,11,19)(8,14,9,13)Polycyclic presentation:
2^7,3|a^2=g,e^2=g,f^2=g,b^a=be,c^a=cf,e^a=eg,f^a=fg,h^a=h^2,c^b=cg,d^b=de,e^b=eg,f^c=fg,h^c=h^2,e^d=eg57 conjugacy classes — view the full character table.
| maximal subgroup of | #10708599, #10708602, #10708604, … |
| maximal normal subgroup of | #10708599, #10708602, #10708604, … |
| maximal quotient of | #10786464, #10786466, #10786510, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 12469);