Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C2×C3×C22⋊C8 ⋊ C2 |
| Sylow | C3 ⋊ #3091 |
420 conjugacy classes of subgroups; only the 137 normal subgroups are shown.
This group has 420 conjugacy classes of subgroups — too many to draw. Only its 137 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)(12,15,17,16)(13,19,20,24)(14,22,23,25)(18,26,27,21);(2,3)(4,6,7,10)(5,9,8,11)(12,14)(13,18)(15,22)(16,25)(17,23)(19,26)(20,27)(21,24);(5,8)(9,11);(2,3)(4,5)(6,9)(7,8)(10,11)(12,13)(14,21)(15,19)(16,24)(17,20)(18,22)(23,26)(25,27)Polycyclic presentation:
2^7,3|c^2=e,d^2=g,f^2=g,b^a=be,c^a=cf,f^a=fg,h^a=h^2,f^c=fg,h^c=h^278 conjugacy classes — view the full character table.
| maximal subgroup of | #10690302, #10690304, #10767346, … |
| maximal normal subgroup of | #10690302, #10690304, #10767346, … |
| maximal quotient of | #10794981, #10795916, #10796404, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 11167);