Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | C4⋊#857 |
| Fitting | C43 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×#857 |
| commutator, cosocle | C2×C42⋊C3 | C22 |
| Frattini, socle | C23 | C2×S4 |
| Fitting | C43 | D3 |
| hypercenter | C4 | #857 |
| hypocenter | C42⋊C3 | D4 |
| LCS | G ▷ C2×C42⋊C3 ▷ C42⋊C3 | |
| UCS | C1 ◁ C2 ◁ C4 | |
164 conjugacy classes of subgroups; only the 15 normal subgroups are shown.
This group has 164 conjugacy classes of subgroups — too many to draw. Only its 15 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 12 |
A minimal faithful permutation representation — 2 generators on 12 points:
(1,9,11,5,7,8,3,6,10,2,4,12);(2,5)(4,8)(6,10)(7,12)(9,11)Polycyclic presentation:
2^3,3,2^4|b^2=c,e^2=g,f^2=gh,b^a=bc,d^a=d^2,e^a=fh,f^a=eh,g^a=gh,e^d=fg,f^d=ef,g^d=gh,h^d=g28 conjugacy classes — view the full character table.
| maximal subgroup of | #11711187, #11711188, #11711230, … |
| maximal normal subgroup of | #11711187, #11711188, #11711230, … |
| maximal quotient of | C4⋊#4621, C4⋊#4622, #11711226, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5558);