Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = C24×S4 (order 384) · Inn(G) = C2×S4 · Out(G) = C23 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C23 |
| unambiguous semidirect product | (C2×S4)⋊C4 |
| Fitting | C23×C4 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C2×S4 |
| commutator | C2×A4 | C2×C4 |
| Fitting | C23×C4 | D3 |
| socle | C24 | C2×D3 |
| prime radical | C22×S4 | C2 |
| cosocle | C22×A4 | C22 |
| hypercenter | C2×C4 | S4 |
| hypocenter | A4 | C22⋊C4 |
| LCS | G ▷ C2×A4 ▷ A4 | |
| UCS | C1 ◁ C22 ◁ C2×C4 | |
143 conjugacy classes of subgroups; only the 23 normal subgroups are shown.
This group has 143 conjugacy classes of subgroups — too many to draw. Only its 23 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 2 generators on 12 points:
(1,2,3,4)(6,8)(9,10,12);(5,6)(7,8)(10,11)Polycyclic presentation:
2^4,3,2^2|b^2=c,b^a=bd,e^a=e^2,f^a=g,g^a=f,f^e=g,g^e=fg28 conjugacy classes — view the full character table.
| maximal subgroup of | #85287, C4×C4⋊S4, #85290, … |
| maximal normal subgroup of | #85287, C4×C4⋊S4, #85290, … |
| maximal quotient of | #73046, #73049, #73052, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 972);