Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #3444⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C24 |
| unambiguous semidirect product | #3444⋊C3 |
| Fitting, Sylow | #3444 ⋊ C3 |
| derived | C22×C42 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C22×C42 | C2×C3 |
| Frattini, socle | C22 | C2×C22⋊A4 |
| Fitting | #3444 | C3 |
| LCS | G ▷ C22×C42 | |
| UCS | C1 | |
232 conjugacy classes of subgroups; only the 17 normal subgroups are shown.
This group has 232 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) = 24 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,6,9,7,10,19)(2,3,14,5,15,20)(4,17,22,18,11,23)(8,12,24,13,16,21);(1,4,5,13)(2,8,7,18)(3,10)(6,15)(9,22,20,21)(11,12)(14,24,19,23)(16,17);(2,7)(8,18)(14,19)(23,24)Polycyclic presentation:
2,3,2^6|e^2=h,f^2=g,b^a=bgh,c^a=cg,d^a=dh,c^b=cdh,d^b=cg,e^b=f,f^b=efgh,g^b=gh,h^b=g24 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#3444⋊C3, #11710660, #62826⋊C3, … |
| maximal normal subgroup of | C2×#3444⋊C3, #11710660, #62826⋊C3, … |
| maximal quotient of | #34012⋊C3, #34551⋊C3, #34552⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18224);