Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) order 3072 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C3×C22⋊Q8 ⋊ C2 |
| Sylow | C3 ⋊ #538 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D3 |
| commutator, socle, cosocle | C22×C3 | C24 |
| Fitting | C3×C22⋊Q8 | C2 |
| hypercenter | C22⋊Q8 | D3 |
| hypocenter | C3 | #538 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C22⋊Q8 | |
250 conjugacy classes of subgroups; only the 95 normal subgroups are shown.
This group has 250 conjugacy classes of subgroups — too many to draw. Only its 95 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 4 generators on 19 points:
(1,2,3)(5,9)(8,11)(12,14,15,19)(13,17,16,18);(6,10)(8,11)(12,13,15,16)(14,18,19,17);(4,6)(5,8)(7,10)(9,11);(2,3)(4,5)(6,8)(7,9)(10,11)(13,16)(17,18)Polycyclic presentation:
2^6,3|c^2=f,d^2=f,c^a=cf,d^a=de,g^a=g^2,c^b=ce,d^c=df36 conjugacy classes — view the full character table.
| maximal subgroup of | #68317, #68319, #68321, … |
| maximal normal subgroup of | #68317, #68319, #68321, … |
| maximal quotient of | #73873, #73928, #73956, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1203);