Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
| direct product | C2×#5426 |
| derived | SL(2,𝔽3) ⋊ C22×C4 |
347 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 347 conjugacy classes of subgroups — too many to draw. Only its 70 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 22 points:
(1,2)(7,13,18,12,9,21,11,19,16,8,17,22)(10,14,15,20);(3,4,5,6)(8,12)(9,10)(13,20)(14,19)(15,17)(16,18);(1,2)52 conjugacy classes — view the full character table.
| maximal subgroup of | #11711307, #11711313, #11711316, … |
| maximal normal subgroup of | #11711307, #11711313, #11711316, … |
| maximal quotient of | C2×#73071, C2×#73072, #11711798, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18094);