Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| Fitting | C3×#533 ⋊ C2 |
| Sylow | C3 ⋊ #3531 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D3 |
| commutator, socle, cosocle | C22×C3 | C25 |
| Frattini | C22 | C24×D3 |
| Fitting | C3×#533 | C2 |
| hypercenter | #533 | D3 |
| hypocenter | C3 | #3531 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C2×C4 ◁ #533 | |
930 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 5 generators on 35 points:
(1,2,3)(20,23,24,31)(21,27,26,33)(22,30,29,35)(25,32,28,34);(5,11)(9,18)(10,14)(17,19)(21,26)(22,29)(27,33)(30,35);(4,7,8,16)(5,10,11,14)(6,12,13,15)(9,17,18,19)(22,29)(25,28)(30,35)(32,34);(4,6,8,13)(5,9,11,18)(7,15,16,12)(10,19,14,17)(20,22)(21,25)(23,30)(24,29)(26,28)(27,32)(31,35)(33,34);(2,3)(4,5)(6,9)(7,14)(8,11)(10,16)(12,19)(13,18)(15,17)(20,21)(22,28)(23,27)(24,26)(25,29)(30,34)(31,33)(32,35)Polycyclic presentation:
2^7,3|b^2=g,c^2=g,e^2=f,b^a=bf,c^a=cg,d^a=dfg,h^a=h^2,c^b=cfg,d^b=df66 conjugacy classes — view the full character table.
| maximal subgroup of | #10687624, #10687627, #10776381, … |
| maximal normal subgroup of | #10687624, #10687627, #10776381, … |
| maximal quotient of | #10957838, #10957843, #10957855, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18862);