Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
| direct product | Q8×Dic12 |
| Sylow | C3 ⋊ Q8×Q16 |
336 conjugacy classes of subgroups; only the 149 normal subgroups are shown.
This group has 336 conjugacy classes of subgroups — too many to draw. Only its 149 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 — 4 generators on 27 points:
(1,2,3)(12,14,21,25,16,23,15,22)(13,17,24,27,19,26,18,20);(2,3)(12,13,16,19)(14,20,23,27)(15,24,21,18)(17,25,26,22);(4,6,7,11)(5,8,9,10);(4,5,7,9)(6,10,11,8)75 conjugacy classes — view the full character table.
| maximal subgroup of | #10708875, #10708879, #10708903, … |
| maximal normal subgroup of | #10708875, #10708879, #10708903, … |
| maximal quotient of | Q8×#816, C4⋊C4×Dic12, Q8×#818, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 12512);