Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C3×C4×D8 ⋊ C2 |
| Sylow | C3 ⋊ #3391 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×D3×D4 |
| commutator, cosocle | C2×C3×C4 | C24 |
| Frattini | C2×C4 | C23×D3 |
| Fitting | C3×C4×D8 | C2 |
| socle | C22×C3 | C22×D4 |
| hypercenter | C4×D8 | D3 |
| hypocenter | C3 | #3391 |
| LCS | G ▷ C2×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C4×D8 | |
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) = 192 |
A minimal faithful permutation representation — 4 generators on 19 points:
(1,2,3)(4,7,8,6)(5,10,9,11);(4,6,8,7)(5,10,9,11)(13,16)(14,19)(17,18);(2,3)(4,5,8,9)(6,10,7,11);(5,9)(10,11)(12,13)(14,16)(15,18)(17,19)Polycyclic presentation:
2^7,3|b^2=e,c^2=e,d^2=e,f^2=g,b^a=be,c^a=cf,f^a=fg,d^b=de,h^b=h^2,f^c=fg63 conjugacy classes — view the full character table.
| maximal subgroup of | #10708694, #10708698, #10708722, … |
| maximal normal subgroup of | #10708694, #10708698, #10708722, … |
| maximal quotient of | #10811974, Dic6×#107, #10814929, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 12154);