Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| Sylow | C3 ⋊ #2665 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, cosocle | C23×C3 | C24 |
| Fitting | C3×#396 | C2 |
| prime radical | C24×C3 | C23 |
| hypercenter | #396 | D3 |
| hypocenter | C3 | #2665 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #396 | |
424 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2)(6,10)(9,11)(12,14,15,18)(13,17,16,19);(4,6)(5,9)(7,10)(8,11)(13,16)(17,19)(20,22,23,27)(21,24,25,26);(2,3)(5,8)(9,11)(20,21,23,25)(22,26,27,24);(4,5)(6,9)(7,8)(10,11)(12,13)(14,17)(15,16)(18,19)Polycyclic presentation:
2^7,3|b^2=g,c^2=g,d^2=f,b^a=be,c^a=cf,c^b=cg,h^b=h^2,d^c=de,h^d=h^254 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#2665, C2×#77346, #10961791, … |
| maximal normal subgroup of | D3×#2665, C2×#77346, #10961791, … |
| maximal quotient of | #10721797, #10721799, #10721804, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 9922);