Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Fitting | C3×C4×C16 ⋊ C2 |
| Sylow | C3 ⋊ #2240 |
160 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 160 conjugacy classes of subgroups — too many to draw. Only its 61 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,3)(5,9)(8,11)(12,14,15,13,20,19,16,24,18,21,27,23,17,25,26,22);(4,6,7,10)(5,8,9,11)(12,13,16,21,17,22,15,19,18,23,26,14,20,24,27,25);(2,3)(4,5)(6,8)(7,9)(10,11)(13,19)(14,24)(15,26)(16,27)(17,20)(21,22)(23,25)Polycyclic presentation:
2^7,3|b^2=e,c^2=d,d^2=fg,e^2=f,f^2=g,b^a=bd,c^a=ce,d^a=df,e^a=efg,f^a=fg,h^a=h^2102 conjugacy classes — view the full character table.
| maximal subgroup of | #10708781, #10708782, #10708788, … |
| maximal normal subgroup of | #10708781, #10708782, #10708788, … |
| maximal quotient of | #10711567, #10713103, #10713401, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 1634);