Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | Aut(G) = C26×D4 (order 512) · Inn(G) = C2×D4 · Out(G) = C25 |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C22 |
| direct product | C3×#463 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C23×C3 |
| prime radical | C2×C3×D8 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C3×C42 ◁ G | |
124 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 124 conjugacy classes of subgroups — too many to draw. Only its 54 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) = 96 |
A minimal faithful permutation representation — 3 generators on 15 points:
(2,4)(5,6)(7,9)(8,10)(11,12)(13,14,15);(1,2,3,4)(6,9)(10,11);(6,10)(8,12)(9,11)57 conjugacy classes — view the full character table.
| Fitting subgroup of | #80504, #80505, #80506, … |
| maximal subgroup of | #80504, #80505, #80506, … |
| maximal normal subgroup of | #80504, #80505, #80506, … |
| maximal quotient of | C3×#1677, C3×#1679, C3×#1681, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 896);