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) = C23 |
| direct product | C3×#452 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C23×C3 |
| prime radical | C2×C3×#41 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C3×C4 ◁ G | |
152 conjugacy classes of subgroups; only the 58 normal subgroups are shown.
This group has 152 conjugacy classes of subgroups — too many to draw. Only its 58 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 — 3 generators on 23 points:
(1,2)(3,4)(5,6,9,13)(7,14,17,20)(8,11,18,12)(10,15,19,16)(21,22,23);(5,7)(6,10)(8,15)(9,17)(11,14)(12,20)(13,19)(16,18);(2,4)(5,8,9,18)(6,11,13,12)(7,15,17,16)(10,14,19,20)57 conjugacy classes — view the full character table.
| Fitting subgroup of | #80212, #80213, #80214, … |
| maximal subgroup of | #80212, #80213, #80214, … |
| maximal normal subgroup of | #80212, #80213, #80214, … |
| maximal quotient of | C3×#1603, C3×#1606, C3×#1615, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 885);