Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C4 | C2×S4 |
| commutator | C2×A4 | C42 |
| Fitting | C22×C42 | D3 |
| socle | C24 | C2×Dic3 |
| prime radical | C22×A4 | C2×C4 |
| cosocle | C2×C4×A4 | C22 |
| hypercenter | C42 | S4 |
| hypocenter | A4 | #97 |
| LCS | G ▷ C2×A4 ▷ A4 | |
| UCS | C1 ◁ C2×C4 ◁ C42 | |
173 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 2 generators on 16 points:
(1,2,3,4)(5,6,8)(10,13)(14,16);(6,7)(9,10,11,14,12,13,15,16)Polycyclic presentation:
2^5,3,2^2|a^2=c,b^2=d,c^2=e,b^a=be,f^a=f^2,g^a=h,h^a=g,g^f=h,h^f=gh56 conjugacy classes — view the full character table.
| maximal subgroup of | #11711021, #11711022, #11711154, … |
| maximal normal subgroup of | #11711021, #11711022, #11711154, … |
| maximal quotient of | #11710985, #11710988, #11710989, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5620);