Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| Sylow | C3 ⋊ #2840 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, cosocle | C23×C3 | C24 |
| Fitting | C3×#379 | C2 |
| prime radical | C25×C3 | C22 |
| hypercenter | #379 | D3 |
| hypocenter | C3 | #2840 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #379 | |
580 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3)(6,10)(8,11)(14,19)(17,18);(5,9)(8,11)(12,14)(13,17)(15,19)(16,18)(20,22,23,27)(21,24,25,26);(2,3)(4,6)(5,8)(7,10)(9,11)(13,16)(17,18)(20,21,23,25)(22,26,27,24);(4,5)(6,8)(7,9)(10,11)(12,13)(14,18)(15,16)(17,19)Polycyclic presentation:
2^7,3|b^2=g,c^2=g,b^a=be,c^a=cf,c^b=cg,d^b=def,h^b=h^2,d^c=de54 conjugacy classes — view the full character table.
| maximal subgroup of | #10698437, #10698438, #10698440, … |
| maximal normal subgroup of | #10698437, #10698438, #10698440, … |
| maximal quotient of | #10725975, #10726315, #10726316, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 6420);