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×C4×Q8 ⋊ C2 |
| Sylow | C3 ⋊ #533 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D3 |
| commutator, socle, cosocle | C22×C3 | C24 |
| Frattini | C22 | C23×D3 |
| Fitting | C3×C4×Q8 | C2 |
| prime radical | C22×D3 | C23 |
| hypercenter | C4×Q8 | D3 |
| hypocenter | C3 | #533 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C2×C4 ◁ C4×Q8 | |
222 conjugacy classes of subgroups; only the 109 normal subgroups are shown.
This group has 222 conjugacy classes of subgroups — too many to draw. Only its 109 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)(5,9)(8,11)(12,14,15,19)(13,16,17,18);(4,6,7,10)(5,8,9,11);(12,13,15,17)(14,18,19,16);(2,3)(4,5)(6,8)(7,9)(10,11)Polycyclic presentation:
2^6,3|b^2=f,c^2=e,d^2=f,d^a=de,g^a=g^2,d^b=df48 conjugacy classes — view the full character table.
| maximal subgroup of | #68232, #68235, #68240, … |
| maximal normal subgroup of | #68232, #68235, #68240, … |
| maximal quotient of | #73618, #73619, C4×#889, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1137);