Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) order 12288 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| Fitting | C2×C3×#41 ⋊ C2 |
| Sylow | C3 ⋊ #535 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D3 |
| commutator, socle, cosocle | C22×C3 | C24 |
| Fitting | C2×C3×#41 | C2 |
| hypercenter | C2×#41 | D3 |
| hypocenter | C3 | #535 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C2×#41 | |
334 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 334 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) = 15 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 4 generators on 15 points:
(1,2,3)(5,7)(9,13)(12,15);(8,10,11,14)(9,12,13,15);(8,9)(10,12)(11,13)(14,15);(2,3)(4,5)(6,7)(10,14)(12,15)Polycyclic presentation:
2^6,3|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 | #70047, #70057, #70061, … |
| maximal normal subgroup of | #70047, #70057, #70061, … |
| maximal quotient of | #82234, #82236, #82257, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1389);