Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) order 6144 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C3×C4×Q8 ⋊ C2 |
| Sylow | C3 ⋊ #556 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D3 |
| commutator, socle, cosocle | C22×C3 | C24 |
| Fitting | C3×C4×Q8 | C2 |
| prime radical | C2×D12 | C22 |
| hypercenter | C4×Q8 | D3 |
| hypocenter | C3 | #556 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C4×Q8 | |
212 conjugacy classes of subgroups; only the 99 normal subgroups are shown.
This group has 212 conjugacy classes of subgroups — too many to draw. Only its 99 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) = 96 |
A minimal faithful permutation representation — 4 generators on 19 points:
(1,2,3)(4,7,8,6)(5,9,10,11)(12,14,15,18)(13,16,17,19);(12,13,15,17)(14,19,18,16);(4,6,8,7)(5,9,10,11);(2,3)(4,5)(6,11)(7,9)(8,10)(14,18)(16,19)Polycyclic presentation:
2^6,3|b^2=f,c^2=e,d^2=ef,b^a=bf,d^a=de,g^a=g^2,d^c=de45 conjugacy classes — view the full character table.
| maximal subgroup of | #68245, #68247, #68250, … |
| maximal normal subgroup of | #68245, #68247, #68250, … |
| maximal quotient of | #73634, #73638, #73643, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1143);