Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C3×C8×Q8 ⋊ C2 |
| Sylow | C3 ⋊ #3361 |
362 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 362 conjugacy classes of subgroups — too many to draw. Only its 123 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 — 4 generators on 27 points:
(1,2,3)(4,6,7,11)(5,9,8,10);(2,3)(13,18)(15,23)(17,24)(19,20)(21,27)(25,26);(4,5,7,8)(6,10,11,9)(12,14,16,22)(13,17,20,26)(15,21,23,27)(18,24,19,25);(2,3)(5,8)(9,10)(12,13,16,20)(14,17,22,26)(15,18,23,19)(21,24,27,25)Polycyclic presentation:
2^7,3|a^2=g,b^2=eg,d^2=e,f^2=g,b^a=be,c^a=cf,f^a=fg,h^a=h^2,d^b=de,f^c=fg,h^c=h^275 conjugacy classes — view the full character table.
| maximal subgroup of | #10785039, #10785046, #10785174, … |
| maximal normal subgroup of | #10785039, #10785046, #10785174, … |
| maximal quotient of | #10811785, #10812003, #10814132, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 12503);