Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C3 ⋊ #1558 |
228 conjugacy classes of subgroups; only the 77 normal subgroups are shown.
This group has 228 conjugacy classes of subgroups — too many to draw. Only its 77 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 — 3 generators on 27 points:
(1,2,3)(13,18)(17,26)(19,23)(20,27);(2,3)(5,8)(6,10)(12,14,16,22)(13,17,18,26)(15,24,25,21)(19,27,23,20);(4,5,6,9,7,8,10,11)(12,13)(14,20)(15,23)(16,18)(17,21)(19,25)(22,27)(24,26)Polycyclic presentation:
2^7,3|a^2=e,b^2=g,d^2=g,e^2=fg,b^a=bd,c^a=cg,d^a=df,d^b=dg,e^b=efg,h^b=h^254 conjugacy classes — view the full character table.
| maximal subgroup of | #10687578, #10687581, #10687713, … |
| maximal normal subgroup of | #10687578, #10687581, #10687713, … |
| maximal quotient of | #10683275, #10683314, #10683421, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 3400);