Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) order 3072 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C3 ⋊ #563 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D3 |
| commutator, socle, cosocle | C22×C3 | C24 |
| Fitting | C3×#123 | C2 |
| prime radical | C2×C3×D4 | C22 |
| hypercenter | #123 | D3 |
| hypocenter | C3 | #563 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ #123 | |
196 conjugacy classes of subgroups; only the 91 normal subgroups are shown.
This group has 196 conjugacy classes of subgroups — too many to draw. Only its 91 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3)(5,9)(8,11)(14,21)(17,26)(20,23)(25,27);(6,10)(8,11)(12,15,16,24)(13,18,19,22)(14,20,21,23)(17,25,26,27);(4,6)(5,8)(7,10)(9,11)(12,14)(13,17)(15,23)(16,21)(18,27)(19,26)(20,24)(22,25);(2,3)(4,5)(6,8)(7,9)(10,11)(12,13,16,19)(14,17,21,26)(15,22,24,18)(20,27,23,25)Polycyclic presentation:
2^6,3|a^2=f,c^2=f,c^a=cf,d^a=def,g^a=g^2,c^b=ce,d^b=df33 conjugacy classes — view the full character table.
| maximal subgroup of | #69141, #71003, C2×#5651, … |
| maximal normal subgroup of | #69141, #71003, C2×#5651, … |
| maximal quotient of | #74062, #74095, #74096, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1210);