Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| isoclinism | isoclinic class of #4621 |
| automorphism | Aut(G) = C2×#857⋊C2 (order 384) · Inn(G) = #857 · Out(G) = C22 |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C1 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #857 |
| commutator, prime radical, cosocle, hypocenter | #95⋊C3 | C2 |
| Frattini, socle | C23 | S4 |
| Fitting | #95 | D3 |
| LCS | G ▷ #95⋊C3 | |
| UCS | C1 ◁ C2 | |
37 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 2 generators on 24 points:
(1,2,14,7,5,8,4,19)(3,21,13,20)(6,15,16,23)(9,11)(10,18,24,17)(12,22);(1,3,10)(2,6,9)(4,11,24)(5,13,15)(7,18,21)(8,17,22)(12,23,14)(16,20,19)Polycyclic presentation:
2,3,2^5|a^2=g,c^2=ef,d^2=e,b^a=b^2,c^a=d,d^a=c,e^a=ef,c^b=dg,d^b=cdf,e^b=fg,f^b=efg,d^c=dg15 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#4621, #72985, #72986, … |
| maximal normal subgroup of | C2×#4621, #72985, #72986, … |
| maximal quotient of | SL(2,ℤ/8), C2×#4621, #10588301, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 180);