Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = D3×#14679 (order 1536) · Inn(G) = C22×D3 · Out(G) = C22×C22⋊C4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C4 |
No aliases for this group.
| Sylow | C3 ⋊ #433 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C4 | C22×D3 |
| commutator, socle | C22×C3 | C22×C4 |
| Fitting | C3×#117 | C2 |
| prime radical | C23×C3 | C2×C4 |
| cosocle | C2×C3×C4 | C23 |
| hypercenter | #117 | D3 |
| hypocenter | C3 | #433 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C2×C4 ◁ #117 | |
96 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,3)(5,9)(8,11)(14,20)(17,23)(21,26)(24,27);(4,6,7,10)(5,8,9,11)(12,14)(13,17)(15,21)(16,20)(18,24)(19,23)(22,26)(25,27);(2,3)(4,5)(6,8)(7,9)(10,11)(12,13,15,18,16,19,22,25)(14,17,21,24,20,23,26,27)Polycyclic presentation:
2^6,3|a^2=d,b^2=e,d^2=f,c^a=ce,g^a=g^2,c^b=cf48 conjugacy classes — view the full character table.
| maximal subgroup of | #78556, #78557, #78558, … |
| maximal normal subgroup of | #78556, #78557, #78558, … |
| maximal quotient of | #69435, #69436, #69437, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 559);