Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) order 12288 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Sylow | C3 ⋊ #536 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D3 |
| commutator, socle, cosocle | C22×C3 | C24 |
| Fitting, prime radical | C2×C3×#41 | C2 |
| hypercenter | C2×#41 | D3 |
| hypocenter | C3 | #536 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C2×#41 | |
270 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 270 conjugacy classes of subgroups — too many to draw. Only its 111 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2,3)(5,7)(10,17)(13,21)(16,23)(20,22);(8,11,12,19)(9,14,15,18)(10,16,17,23)(13,20,21,22);(8,10)(9,13)(11,16)(12,17)(14,20)(15,21)(18,22)(19,23);(2,3)(4,5)(6,7)(8,9,12,15)(10,13,17,21)(11,18,19,14)(16,22,23,20)Polycyclic presentation:
2^6,3|a^2=f,c^2=f,c^a=cf,d^a=de,g^a=g^2,d^b=df42 conjugacy classes — view the full character table.
| maximal subgroup of | #70046, #70058, #70059, … |
| maximal normal subgroup of | #70046, #70058, #70059, … |
| maximal quotient of | #82230, #82234, #82235, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1384);