Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C3 ⋊ #1467 |
236 conjugacy classes of subgroups; only the 89 normal subgroups are shown.
This group has 236 conjugacy classes of subgroups — too many to draw. Only its 89 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)(4,6,8,7)(5,9,10,11)(12,15,16,23)(13,18,19,25)(14,21,22,27)(17,24,20,26);(12,14,16,22)(13,17,19,20)(15,21,23,27)(18,24,25,26);(2,3)(4,5,7,9,8,10,6,11)(12,13)(14,20,22,17)(15,18)(16,19)(21,26,27,24)(23,25)Polycyclic presentation:
2^7,3|a^2=e,b^2=f,c^2=fg,e^2=g,b^a=bd,c^a=cg,d^a=df,h^a=h^2,e^b=ef72 conjugacy classes — view the full character table.
| maximal subgroup of | #10682302, #10682303, #10682501, … |
| maximal normal subgroup of | #10682302, #10682303, #10682501, … |
| maximal quotient of | #10683380, #10683393, #10683406, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 4201);