Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| Fitting | C3×#541 ⋊ C2 |
| Sylow | C3 ⋊ #3566 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D3 |
| commutator, socle, cosocle | C22×C3 | C25 |
| Fitting | C3×#541 | C2 |
| hypercenter | #541 | D3 |
| hypocenter | C3 | #3566 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ #541 | |
954 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 5 generators on 35 points:
(1,2)(5,11)(9,12)(10,17)(16,18)(20,23,24,30)(21,27,26,33)(22,29,28,34)(25,32,31,35);(2,3)(6,14)(9,12)(13,19)(16,18)(21,26)(25,31)(27,33)(32,35);(2,3)(4,7,8,15)(5,10,11,17)(6,13,14,19)(9,16,12,18)(21,26)(22,28)(27,33)(29,34);(2,3)(4,6)(5,9)(7,13)(8,14)(10,16)(11,12)(15,19)(17,18)(20,22)(21,25)(23,29)(24,28)(26,31)(27,32)(30,34)(33,35);(2,3)(4,5)(6,12)(7,10)(8,11)(9,14)(13,18)(15,17)(16,19)(20,21)(22,25)(23,27)(24,26)(28,31)(29,32)(30,33)(34,35)Polycyclic presentation:
2^7,3|c^2=f,e^2=g,b^a=bf,c^a=cg,d^a=dg,e^a=ef,h^a=h^2,c^b=cg,d^b=df,h^b=h^2,h^c=h^2,h^d=h^2,h^e=h^260 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#86507, #11702751, #11702765, … |
| maximal normal subgroup of | D3×#3566, C2×#86507, #11702751, … |
| maximal quotient of | #10958269, #10958384, #10958457, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 19083);