Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| Fitting | C3×#556 ⋊ C2 |
| Sylow | C3 ⋊ #3565 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D3 |
| commutator, socle, cosocle | C22×C3 | C25 |
| Fitting | C3×#556 | C2 |
| hypercenter | #556 | D3 |
| hypocenter | C3 | #3565 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ #556 | |
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)(20,24,25,23)(21,29,28,27)(22,31,32,33)(26,34,30,35);(2,3)(4,7,8,15)(5,11,10,14)(6,13,12,18)(9,16,17,19)(21,28)(22,32)(27,29)(31,33);(5,10)(6,12)(11,14)(13,18)(20,23,25,24)(21,27,28,29)(22,31,32,33)(26,34,30,35);(4,6)(5,9)(7,13)(8,12)(10,17)(11,16)(14,19)(15,18)(20,22)(21,26)(23,33)(24,31)(25,32)(27,35)(28,30)(29,34);(4,5,8,10)(6,9,12,17)(7,14,15,11)(13,19,18,16)(20,21,25,28)(22,30,32,26)(23,29,24,27)(31,34,33,35)Polycyclic presentation:
2^7,3|a^2=g,c^2=f,d^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=df,h^d=h^263 conjugacy classes — view the full character table.
| maximal subgroup of | #11692003, #11692007, #11692018, … |
| maximal normal subgroup of | #11692003, #11692007, #11692018, … |
| maximal quotient of | #10958234, #10958340, #10958493, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 19576);