Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = C2×D3×D42 (order 768) · Inn(G) = #884 · Out(G) = C23 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C3×#100 ⋊ C2 |
| Sylow | C3 ⋊ #472 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #884 |
| commutator, cosocle | C2×C3×C4 | C23 |
| Frattini | C2×C4 | C22×D3 |
| Fitting | C3×#100 | C2 |
| socle | C2×C3 | #121 |
| hypercenter | #100 | D3 |
| hypocenter | C3 | #472 |
| LCS | G ▷ C2×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #100 | |
108 conjugacy classes of subgroups; only the 35 normal subgroups are shown.
This group has 108 conjugacy classes of subgroups — too many to draw. Only its 35 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 — 3 generators on 19 points:
(1,2,3)(5,10)(6,13)(7,15)(9,16)(11,12)(17,18);(4,6,7,14,8,13,15,19)(5,9,11,17,12,18,10,16);(2,3)(4,5)(6,9)(7,11)(8,12)(10,15)(13,18)(14,17)(16,19)Polycyclic presentation:
2^6,3|b^2=d,d^2=f,c^a=cde,e^a=ef,g^a=g^2,c^b=cef,e^b=ef,d^c=df27 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#4749, #78703, #78719, … |
| maximal normal subgroup of | C2×#4749, #78703, #78719, … |
| maximal quotient of | #68395, #68396, #68399, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 308);