Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C3×#401 ⋊ C2 |
| Sylow | C3 ⋊ #2838 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, cosocle | C23×C3 | C24 |
| Fitting | C3×#401 | C2 |
| prime radical | C23×D3 | C23 |
| hypercenter | #401 | D3 |
| hypocenter | C3 | #2838 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #401 | |
366 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)(16,19)(20,22,23,27)(21,25,24,26);(6,10)(8,11)(12,14,15,18)(13,16,17,19)(22,27)(25,26);(4,6)(5,8)(7,10)(9,11)(12,13)(14,16)(15,17)(18,19)(20,21,23,24)(22,26,27,25);(2,3)(4,5,7,9)(6,8,10,11)(14,18)(16,19)(21,24)(25,26)Polycyclic presentation:
2^7,3|a^2=g,b^2=e,c^2=f,d^2=e,b^a=be,c^a=cf,h^a=h^2,c^b=cg,d^b=def,d^c=de,h^d=h^248 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#2838, C2×#78428, #11001486, … |
| maximal normal subgroup of | D3×#2838, C2×#78428, #11001486, … |
| maximal quotient of | #10725914, #10725922, #10725964, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 11004);