Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = C2×D4×S4 (order 384) · Inn(G) = C2×S4 · Out(G) = C23 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C4 | C2×S4 |
| commutator | C2×A4 | C2×C4 |
| Fitting | C23×C4 | D3 |
| socle | C23 | C2×Dic3 |
| prime radical | C22×A4 | C4 |
| cosocle | C4×A4 | C22 |
| hypercenter | C2×C4 | S4 |
| hypocenter | A4 | M16 |
| LCS | G ▷ C2×A4 ▷ A4 | |
| UCS | C1 ◁ C4 ◁ C2×C4 | |
81 conjugacy classes of subgroups.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 2 generators on 12 points:
(1,2,4)(6,9)(10,12);(2,3)(5,6,7,10,8,9,11,12)Polycyclic presentation:
2^4,3,2^2|a^2=c,c^2=d,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 | #73046, #73105, #73114, … |
| maximal normal subgroup of | #73046, #73105, #73114, … |
| maximal quotient of | #73044, C4⋊(A4⋊C8), #73055, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 968);