Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = C2×D4×S4 (order 384) · Inn(G) = GL(2,ℤ/4) · Out(G) = C22 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2 | GL(2,ℤ/4) |
| commutator, cosocle | C4×A4 | C22 |
| Frattini | C4 | C2×S4 |
| Fitting | C22×Q8 | D3 |
| socle | C23 | #67 |
| prime radical | C2×A4 | D4 |
| hypercenter | Q8 | S4 |
| hypocenter | A4 | Q16 |
| LCS | G ▷ C4×A4 ▷ C2×A4 ▷ A4 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ Q8 | |
70 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 2 generators on 20 points:
(1,2,4)(5,7,9,15)(6,10,12,19)(8,17,18,14)(11,20,16,13);(2,3)(5,6,9,12)(7,13,15,20)(8,16,18,11)(10,17,19,14)Polycyclic presentation:
2^4,3,2^2|a^2=d,b^2=d,c^2=d,b^a=bc,c^a=cd,e^a=e^2,f^a=g,g^a=f,c^b=cd,f^e=g,g^e=fg19 conjugacy classes — view the full character table.
| maximal subgroup of | #85461, #85462, Q16×S4, … |
| maximal normal subgroup of | #85461, #85462, Q16×S4, … |
| maximal quotient of | #73050, #73052, #73093, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 975);