Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Sylow | C3 ⋊ #1768 |
260 conjugacy classes of subgroups; only the 129 normal subgroups are shown.
This group has 260 conjugacy classes of subgroups — too many to draw. Only its 129 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) = 384 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,3)(4,6,7,5)(13,18)(17,23)(19,25)(24,27);(9,11)(12,14,15,20,16,21,22,26)(13,17,19,24,18,23,25,27);(2,3)(4,5,7,6)(8,9,10,11)(12,13)(14,17)(15,19)(16,18)(20,24)(21,23)(22,25)(26,27)Polycyclic presentation:
2^7,3|a^2=e,b^2=f,c^2=de,f^2=g,b^a=bd,c^a=cg,h^a=h^2120 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#1768, C2×#69755, #10793074, … |
| maximal normal subgroup of | D3×#1768, C2×#69755, #10793074, … |
| maximal quotient of | #10686333, #10686340, #10686363, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 2331);