Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | Aut(G) = C22×C4×S4 (order 384) · Inn(G) = C2×S4 · Out(G) = C2×C4 |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting | C22×C16 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C8 | C2×S4 |
| commutator | C2×A4 | C2×C8 |
| Fitting | C22×C16 | D3 |
| socle | C23 | D3×C8 |
| prime radical | C2×S4 | C8 |
| cosocle | C8×A4 | C22 |
| hypercenter | C16 | S4 |
| hypocenter | A4 | M32 |
| LCS | G ▷ C2×A4 ▷ A4 | |
| UCS | C1 ◁ C8 ◁ C16 | |
95 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 2 generators on 20 points:
(1,2,4)(5,6,7,11,8,12,13,18,9,10,14,16,15,17,19,20);(2,3)(6,10)(11,16)(12,17)(18,20)Polycyclic presentation:
2^5,3,2^2|b^2=c,c^2=d,d^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 | C2×#73034, #11714132, #11714133, … |
| maximal normal subgroup of | C2×#73034, #11714132, #11714133, … |
| maximal quotient of | #11711379, #11711381, #11711383, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5610);