Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #67992 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting | C82 ⋊ D3 |
| derived | C82⋊C3 ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C82⋊C3 | C2 |
| Frattini | C42 | S4 |
| Fitting | C82 | D3 |
| socle | C22 | #857 |
| LCS | G ▷ C82⋊C3 | |
| UCS | C1 | |
47 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 2 generators on 24 points:
(1,3,5,4,6,11,13,14)(2,23,22,24,18,15,20,16,10,21,12,17,8,19,9,7);(1,2,7)(3,9,19)(4,12,21)(5,8,17)(6,10,16)(11,20,15)(13,18,24)(14,22,23)Polycyclic presentation:
2,3,2^6|c^2=e,d^2=efh,e^2=g,f^2=h,b^a=b^2,c^a=deg,d^a=cefgh,e^a=efgh,g^a=gh,c^b=deh,d^b=cd,e^b=efgh,f^b=e,g^b=gh,h^b=g24 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#67992, #11711355, #11711452, … |
| maximal normal subgroup of | C2×#67992, #11711355, #11711452, … |
| maximal quotient of | #11710948, #11710949, #11710954, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 568);