Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #4621 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| direct product | C2×#4621 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | #857 |
| commutator, cosocle, hypocenter | #95⋊C3 | C22 |
| Frattini | C23 | C2×S4 |
| Fitting | C2×#95 | D3 |
| socle | C24 | S4 |
| prime radical | C2×#95⋊C3 | C2 |
| LCS | G ▷ #95⋊C3 | |
| UCS | C1 ◁ C22 | |
138 conjugacy classes of subgroups; only the 17 normal subgroups are shown.
This group has 138 conjugacy classes of subgroups — too many to draw. Only its 17 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 2 generators on 26 points:
(3,9)(4,6,7,12)(5,14,15,13,20,25,10,26)(16,22,19,23,17,21,18,24);(1,2)(3,5,14)(4,10,13)(6,18,26)(7,19,24)(8,20,22)(9,17,21)(11,16,25)(12,15,23)30 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#4621, #11711006, #11711089, … |
| maximal normal subgroup of | C4×#4621, #11711006, #11711089, … |
| maximal quotient of | C2×SL(2,ℤ/8), C4×#4621, #11711160, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5559);