Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | Aut(G) = C2×#1202 (order 2048) · Inn(G) = C24 · Out(G) = C2×D42 |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#547 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C24 |
| commutator, Frattini, cosocle | C22 | C24×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ G | |
292 conjugacy classes of subgroups; only the 166 normal subgroups are shown.
This group has 292 conjugacy classes of subgroups — too many to draw. Only its 166 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 — 4 generators on 15 points:
(1,2)(3,4)(5,6,9,8)(7,11,12,10)(13,14,15);(5,7)(6,10)(8,11)(9,12);(2,4)(5,8,9,6)(7,11,12,10);(7,12)(10,11)75 conjugacy classes — view the full character table.
| Fitting subgroup of | #71910, #71911, #71912, … |
| maximal subgroup of | #71910, #71911, #71912, … |
| maximal normal subgroup of | #71910, #71911, #71912, … |
| maximal quotient of | C3×#2339, C3×#2344, C3×D4×C4⋊C4, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1436);