Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C3 ⋊ #1288 |
152 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 152 conjugacy classes of subgroups — too many to draw. Only its 57 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 — 2 generators on 27 points:
(1,2)(5,8,10,9)(12,14,15,22,16,21,23,27)(13,17,19,24,18,20,25,26);(2,3)(4,5)(6,8)(7,10)(9,11)(12,13)(14,20,21,17)(15,19)(16,18)(22,26,27,24)(23,25)Polycyclic presentation:
2^7,3|a^2=d,b^2=e,c^2=fg,e^2=g,b^a=bc,c^a=cf,e^a=efg,h^a=h^2,d^b=dg,h^b=h^260 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#1288, C2×#67752, #10685029, … |
| maximal normal subgroup of | D3×#1288, C2×#67752, #10685029, … |
| maximal quotient of | #10681722, #10681734, #10681735, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 328);