Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| Sylow | C3 ⋊ #3211 |
410 conjugacy classes of subgroups; only the 113 normal subgroups are shown.
This group has 410 conjugacy classes of subgroups — too many to draw. Only its 113 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) = 96 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3)(5,9)(8,10)(13,19)(17,21)(18,20)(25,26);(6,11)(8,10)(13,18)(14,22)(16,24)(17,25)(19,20)(21,26);(2,3)(4,6,7,11)(5,8,9,10)(12,14)(13,17)(15,22)(16,23)(18,26)(19,21)(20,25)(24,27);(4,5)(6,10)(7,9)(8,11)(12,13)(14,21)(15,19)(16,18)(17,22)(20,24)(23,25)(26,27)Polycyclic presentation:
2^7,3|b^2=eg,f^2=g,b^a=be,c^a=cf,d^a=de,f^a=fg,c^b=ce,h^b=h^2,f^c=fg51 conjugacy classes — view the full character table.
| maximal subgroup of | #10769200, #10769202, #10769211, … |
| maximal normal subgroup of | #10769200, #10769202, #10769211, … |
| maximal quotient of | #10804509, #10804873, #10805714, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 12571);