Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C8 | C2×S4 |
| commutator | C2×A4 | C2×C8 |
| Fitting | C23×C8 | D3 |
| socle | C23 | C2×C3⋊C8 |
| prime radical | C22×A4 | C8 |
| cosocle | C8×A4 | C22 |
| hypercenter | C2×C8 | S4 |
| hypocenter | A4 | M32 |
| LCS | G ▷ C2×A4 ▷ A4 | |
| UCS | C1 ◁ C8 ◁ C2×C8 | |
107 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)(6,10)(11,16)(12,17)(18,20);(2,3)(5,6,7,11,8,12,13,18,9,10,14,16,15,17,19,20)Polycyclic presentation:
2^5,3,2^2|a^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 | #11711484, #11711490, #11711495, … |
| maximal normal subgroup of | #11711484, #11711490, #11711495, … |
| maximal quotient of | #11711348, #11711350, #11711352, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5680);