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.
| Fitting | C3×#117 ⋊ C2 |
| Sylow | C3 ⋊ #536 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D3 |
| commutator, socle, cosocle | C22×C3 | C24 |
| Fitting | C3×#117 | C2 |
| prime radical | C2×#235 | C2 |
| hypercenter | #117 | D3 |
| hypocenter | C3 | #536 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ #117 | |
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)(4,5,6,7)(9,15)(10,17)(11,19)(20,22);(8,11)(9,14)(10,16)(12,19)(13,20)(15,18)(17,23)(21,22);(8,10,12,17)(9,13,15,21)(11,16,19,23)(14,20,18,22);(2,3)(5,7)(8,9)(10,13)(11,18)(12,15)(14,19)(16,22)(17,21)(20,23)Polycyclic presentation:
2^6,3|b^2=f,d^2=ef,c^a=cf,d^a=de,g^a=g^2,d^b=df,d^c=df42 conjugacy classes — view the full character table.
| maximal subgroup of | #68708, #68712, #68734, … |
| maximal normal subgroup of | #68708, #68712, #68734, … |
| maximal quotient of | #73387, #73388, #73403, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1085);