Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Sylow | C3 ⋊ #3407 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C8 | C23×D3 |
| commutator, socle | C2×C3 | C23×C8 |
| Fitting | C2×C3×#131 | C2 |
| prime radical | C2×C3×#41 | C4 |
| cosocle | C3×C8 | C24 |
| hypercenter | C2×#131 | D3 |
| hypocenter | C3 | #3407 |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C8 ◁ C2×#131 | |
360 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 35 points:
(1,2,3)(5,11)(10,20)(12,23)(13,24)(21,30)(22,31)(25,33)(32,35);(6,14)(10,20)(15,26)(16,27)(21,30)(22,31)(28,34)(32,35);(4,6)(5,10)(7,15)(8,16)(9,14)(11,20)(12,21)(13,22)(17,28)(18,26)(19,27)(23,30)(24,31)(25,32)(29,34)(33,35);(2,3)(4,5,7,12,8,13,17,25,9,11,18,23,19,24,29,33)(6,10,15,21,16,22,28,32,14,20,26,30,27,31,34,35)Polycyclic presentation:
2^7,3|a^2=e,e^2=f,f^2=g,d^a=dg,h^a=h^2,c^b=cg108 conjugacy classes — view the full character table.
| maximal subgroup of | #10710315, #10710317, #10711346, … |
| maximal normal subgroup of | #10710315, #10710317, #10711346, … |
| maximal quotient of | #10947828, #10947837, #10947865, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 16356);