Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C4 | C22×S4 |
| commutator | C2×A4 | C22×C4 |
| Fitting | C22×#41 | D3 |
| socle | C23 | C22×Dic3 |
| prime radical | A4×#41 | C2 |
| cosocle | C4×A4 | C23 |
| hypercenter | #41 | S4 |
| hypocenter | A4 | #131 |
| LCS | G ▷ C2×A4 ▷ A4 | |
| UCS | C1 ◁ C4 ◁ #41 | |
314 conjugacy classes of subgroups; only the 62 normal subgroups are shown.
This group has 314 conjugacy classes of subgroups — too many to draw. Only its 62 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2,4)(7,13)(10,16)(14,19)(17,20);(5,7)(6,10)(8,14)(9,13)(11,17)(12,16)(15,19)(18,20);(2,3)(5,6,8,11,9,12,15,18)(7,10,14,17,13,16,19,20)Polycyclic presentation:
2^5,3,2^2|a^2=d,d^2=e,f^a=f^2,g^a=h,h^a=g,c^b=ce,g^f=h,h^f=gh50 conjugacy classes — view the full character table.
| maximal subgroup of | #11712942, #11712945, #11713080, … |
| maximal normal subgroup of | #11712942, #11712945, #11713080, … |
| maximal quotient of | #11711803, #11711804, #11711805, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18087);