Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
| direct product | C4×#836 |
| Sylow | C3 ⋊ C4×#101 |
296 conjugacy classes of subgroups; only the 129 normal subgroups are shown.
This group has 296 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) = 23 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2,3)(8,10,12,18)(9,13,15,16)(11,19,20,17)(14,22,23,21);(2,3)(8,9,11,14,12,15,20,23)(10,16,17,22,18,13,19,21);(4,5,6,7)84 conjugacy classes — view the full character table.
| maximal subgroup of | #10682190, #10682191, #10682401, … |
| maximal normal subgroup of | #10682190, #10682191, #10682401, … |
| maximal quotient of | C4×#4528, #10685701, #10685705, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 4212);