Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = C28×D3 (order 1536) · Inn(G) = C23×D3 · Out(G) = C25 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C3 ⋊ #541 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D3 |
| commutator, socle, cosocle | C22×C3 | C24 |
| Fitting | C3×#126 | C2 |
| prime radical | C23×C3 | C23 |
| hypercenter | #126 | D3 |
| hypocenter | C3 | #541 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ #126 | |
192 conjugacy classes of subgroups; only the 93 normal subgroups are shown.
This group has 192 conjugacy classes of subgroups — too many to draw. Only its 93 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3)(6,11)(9,10)(12,15,16,24)(13,18,19,22)(14,21,20,23)(17,26,25,27);(4,7,8,5)(6,10,11,9)(14,20)(17,25)(21,23)(26,27);(4,6)(5,9)(7,10)(8,11)(12,14,16,20)(13,17,19,25)(15,23,24,21)(18,27,22,26);(2,3)(4,5,8,7)(6,9,11,10)(12,13,16,19)(14,17,20,25)(15,22,24,18)(21,27,23,26)Polycyclic presentation:
2^6,3|a^2=f,b^2=e,c^2=ef,d^2=e,d^a=de,g^a=g^2,c^b=ce,d^b=df36 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#5701, #86110, #86116, … |
| maximal normal subgroup of | C2×#5701, #86110, #86116, … |
| maximal quotient of | #74260, #74261, #74267, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1260);