Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C3 ⋊ #3123 |
394 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 394 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) = 96 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3)(4,6,7,10)(5,9,8,11)(14,22)(17,21)(23,27)(25,26);(13,18)(16,24)(17,25)(19,20)(21,26)(23,27);(5,8)(9,11)(12,14)(13,17)(15,22)(16,23)(18,25)(19,21)(20,26)(24,27);(2,3)(4,5)(6,9)(7,8)(10,11)(12,13,15,19)(14,21,22,17)(16,18,24,20)(23,26,27,25)Polycyclic presentation:
2^7,3|a^2=g,d^2=eg,f^2=g,b^a=be,c^a=cf,f^a=fg,h^a=h^2,d^b=dg,f^c=fg66 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#3123, C2×#82545, #11663548, … |
| maximal normal subgroup of | D3×#3123, C2×#82545, #11663548, … |
| maximal quotient of | #10795278, #10798263, #10798367, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 15121);