Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting, Sylow | #2840 ⋊ C3 |
| derived | C22×Q8 ⋊ C22×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #5450 |
| commutator, cosocle, hypocenter | C22×Q8 | C22×C3 |
| Frattini, socle | C23 | C22×A4 |
| Fitting | #2840 | C3 |
| LCS | G ▷ C22×Q8 | |
| UCS | C1 ◁ C2 | |
160 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 2 generators on 24 points:
(1,2,6)(3,22,23,10,11,16)(4,7,19)(5,9,17)(8,15,12,20,21,14)(13,18,24);(1,12)(2,22,9,11)(3,13)(4,10)(5,14)(6,19)(7,20,18,8)(15,16)(17,24)(21,23)Polycyclic presentation:
2^2,3,2^5|d^2=h,e^2=h,d^a=df,e^a=eg,d^b=dfg,e^b=ef,d^c=defh,e^c=dfg,f^c=fg,g^c=f,e^d=eh30 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#73210, #11710042, #52925⋊C3, … |
| maximal normal subgroup of | C2×#73210, #11710042, #52925⋊C3, … |
| maximal quotient of | #14635⋊C3, #17216⋊C3, #11709342, … |
| automorphism group of | #2032⋊C3 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5786);