Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C3 ⋊ #1649 |
234 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 234 conjugacy classes of subgroups — too many to draw. Only its 59 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)(12,15,18,14)(13,19,22,27)(16,24,17,25)(20,26,21,23);(2,3)(5,8)(6,11)(9,10)(12,14,18,15)(13,19,22,27)(16,25,17,24)(20,26,21,23);(2,3)(4,5)(6,8)(7,10)(9,11)(12,13,18,22)(14,23,15,26)(16,21,17,20)(19,25,27,24)Polycyclic presentation:
2^7,3|a^2=g,b^2=g,c^2=g,d^2=f,e^2=g,b^a=bd,c^a=ce,d^a=df,e^a=eg,h^a=h^2,d^b=df,e^b=eg,h^b=h^2,d^c=dg,e^c=eg,h^c=h^254 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#1649, C2×#68192, #10771023, … |
| maximal normal subgroup of | D3×#1649, C2×#68192, #10771023, … |
| maximal quotient of | #10684011, #10684017, #10684053, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 768);