Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| Fitting | C3×#394 ⋊ C2 |
| Sylow | C3 ⋊ #2756 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, cosocle | C23×C3 | C24 |
| Fitting | C3×#394 | C2 |
| prime radical | #5223 | C2 |
| hypercenter | #394 | D3 |
| hypocenter | C3 | #2756 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #394 | |
484 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)(13,17)(16,18)(20,22,23,26)(21,25,24,27);(2,3)(6,10)(8,11)(12,14,15,19)(13,16,17,18)(21,24)(25,27);(2,3)(4,6)(5,8)(7,10)(9,11)(21,24)(22,26);(4,5,7,9)(6,8,10,11)(12,13)(14,18)(15,17)(16,19)(20,21)(22,25)(23,24)(26,27)Polycyclic presentation:
2^7,3|a^2=g,c^2=ef,d^2=e,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,h^b=h^2,h^c=h^2,h^d=h^254 conjugacy classes — view the full character table.
| maximal subgroup of | #11003045, #11003066, #11003284, … |
| maximal normal subgroup of | #11003045, #11003066, #11003284, … |
| maximal quotient of | #10722780, #10723041, #10723053, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 9637);