Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| Fitting | C3×C4×C4⋊C4 ⋊ C2 |
| Sylow | C3 ⋊ #2491 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, cosocle | C23×C3 | C24 |
| Fitting | C3×C4×C4⋊C4 | C2 |
| prime radical | C2×#895 | C2 |
| hypercenter | C4×C4⋊C4 | D3 |
| hypocenter | C3 | #2491 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊C4 | |
632 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 19 points:
(1,2,3)(4,6,7,5);(2,3)(8,9)(10,11)(13,17)(16,19);(4,5,7,6)(8,9,10,11)(12,14,15,18)(13,16,17,19);(5,6)(8,9,10,11)(12,13)(14,16)(15,17)(18,19)Polycyclic presentation:
2^7,3|a^2=g,b^2=ef,d^2=e,b^a=be,c^a=cf,d^a=de,c^b=cg,h^c=h^278 conjugacy classes — view the full character table.
| maximal subgroup of | #10863218, #10863222, #10863278, … |
| maximal normal subgroup of | #10863218, #10863222, #10863278, … |
| maximal quotient of | #10717132, #10717541, #10718282, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 6364);