Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C3 ⋊ #3340 |
352 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 352 conjugacy classes of subgroups — too many to draw. Only its 123 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)(6,11)(9,10)(12,14,17,16)(13,18,20,24)(15,21,23,25)(19,26,27,22);(4,7,8,5)(6,10,11,9)(12,15)(13,19)(14,21)(16,25)(17,23)(18,26)(20,27)(22,24);(4,6)(5,9)(7,10)(8,11)(12,14,17,16)(13,18,20,24)(15,21,23,25)(19,26,27,22);(2,3)(4,5,8,7)(6,10,11,9)(12,13)(14,18)(15,22)(16,24)(17,20)(19,21)(23,26)(25,27)Polycyclic presentation:
2^7,3|a^2=e,b^2=g,c^2=e,d^2=g,f^2=g,b^a=be,c^a=cf,f^a=fg,h^a=h^2,d^b=de,f^c=fg60 conjugacy classes — view the full character table.
| maximal subgroup of | #10785067, #10785074, #10785202, … |
| maximal normal subgroup of | #10785067, #10785074, #10785202, … |
| maximal quotient of | #10811453, #10811458, #10812823, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 16222);