Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| isoclinism | isoclinic class of #4656 |
| automorphism | Aut(G) order 6144 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C3 ⋊ #398 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C22×D3 |
| commutator, socle, prime radical, cosocle | C23×C3 | C23 |
| Fitting | C2×C3×C4⋊C4 | C2 |
| hypercenter | C2×C4⋊C4 | D3 |
| hypocenter | C3 | #398 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C2×C4⋊C4 | |
150 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,2,3)(5,7)(8,9)(10,11)(12,14,15,18)(13,17,16,19);(8,9,10,11)(13,16)(17,19);(2,3)(4,5)(6,7)(8,9,10,11)(12,13)(14,17)(15,16)(18,19)Polycyclic presentation:
2^6,3|a^2=f,b^2=f,c^2=e,b^a=be,c^a=cd,g^a=g^2,c^b=cf42 conjugacy classes — view the full character table.
| maximal subgroup of | #71166, #71167, #71174, … |
| maximal normal subgroup of | #71166, #71167, #71174, … |
| maximal quotient of | #69409, #69413, #69541, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 542);