Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| isoclinism | isoclinic class of Dic6 |
| automorphism | Aut(G) = C2×D3×#525 (order 768) · Inn(G) = C2×D3 · Out(G) = C22×#41 |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C3 ⋊ #346 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C8 | C2×D3 |
| commutator | C2×C3 | C4×C8 |
| Fitting | C2×C3×C16 | C2 |
| socle, prime radical | C22×C3 | C2×C8 |
| cosocle | C2×C3×C8 | C22 |
| hypercenter | C2×C16 | D3 |
| hypocenter | C3 | #346 |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2×C8 ◁ C2×C16 | |
54 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 2 generators on 23 points:
(1,2,3)(8,9,10,14,11,15,16,21,12,13,17,19,18,20,22,23);(2,3)(4,5,6,7)(9,13)(14,19)(15,20)(21,23)Polycyclic presentation:
2^6,3|a^2=c,b^2=d,d^2=e,e^2=f,b^a=bf,g^a=g^272 conjugacy classes — view the full character table.
| maximal subgroup of | #67600, #69048, C4×#798, … |
| maximal normal subgroup of | #67600, #69048, C4×#798, … |
| maximal quotient of | #67498, #67499, #67600, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 61);