Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C2×C3×C4⋊Q8 ⋊ C2 |
| Sylow | C3 ⋊ #3238 |
362 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 362 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) = 27 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3)(13,19)(17,20)(18,27)(25,26);(5,10)(6,7)(9,11)(12,15,16,24)(13,18,19,27)(14,21,22,23)(17,25,20,26);(4,6,8,7)(5,9,11,10)(12,14,16,22)(13,17,19,20)(15,23,24,21)(18,26,27,25);(2,3)(4,5)(6,9)(7,10)(8,11)(12,13)(14,20)(15,18)(16,19)(17,22)(21,26)(23,25)(24,27)Polycyclic presentation:
2^7,3|b^2=eg,c^2=e,f^2=g,b^a=be,c^a=cf,d^a=de,f^a=fg,h^a=h^2,c^b=ceg,f^c=fg54 conjugacy classes — view the full character table.
| maximal subgroup of | #10692184, #10692190, #10692443, … |
| maximal normal subgroup of | #10692184, #10692190, #10692443, … |
| maximal quotient of | #10804716, #10804927, #10805751, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 15793);