Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C3×#474 ⋊ C2 |
| Sylow | C3 ⋊ Q8×D8 |
404 conjugacy classes of subgroups; only the 133 normal subgroups are shown.
This group has 404 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) = 192 |
A minimal faithful permutation representation — 4 generators on 19 points:
(1,2,3)(4,7,8,5)(6,9,11,10);(13,16)(14,19)(17,18);(2,3)(4,6,8,11)(5,9,7,10);(2,3)(4,5,8,7)(6,10,11,9)(12,13)(14,16)(15,18)(17,19)Polycyclic presentation:
2^7,3|a^2=e,b^2=e,d^2=e,f^2=g,b^a=be,c^a=cf,f^a=fg,h^a=h^2,d^b=de,h^b=h^2,f^c=fg54 conjugacy classes — view the full character table.
| maximal subgroup of | #10784561, #10784562, #10784568, … |
| maximal normal subgroup of | #10784561, #10784562, #10784568, … |
| maximal quotient of | #10811771, #10811775, #10813575, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 13533);