Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C3×Q8⋊C8 ⋊ C2 |
| Sylow | C3 ⋊ #1495 |
216 conjugacy classes of subgroups; only the 65 normal subgroups are shown.
This group has 216 conjugacy classes of subgroups — too many to draw. Only its 65 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)(5,9)(8,11)(13,18)(14,20)(19,25)(21,26);(2,3)(4,5)(6,11)(7,9)(8,10)(12,14)(13,17)(15,21)(16,20)(18,24)(19,23)(22,26)(25,27);(5,8)(6,10)(9,11)(12,13,15,19,16,18,22,25)(14,17,21,23,20,24,26,27)Polycyclic presentation:
2^7,3|a^2=e,d^2=f,e^2=g,b^a=bd,c^a=cg,d^a=df,c^b=cfg,d^b=df,h^b=h^2,h^c=h^260 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#1495, C2×#68242, #10761820, … |
| maximal normal subgroup of | D3×#1495, C2×#68242, #10761820, … |
| maximal quotient of | #10683702, #10683707, #10683712, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 818);