Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = C25×D3×D4 (order 1536) · Inn(G) = D3×D4 · Out(G) = C25 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C2×C3×D8 ⋊ C2 |
| Sylow | C3 ⋊ #459 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | D3×D4 |
| commutator, cosocle | C2×C3×C4 | C23 |
| Frattini | C2×C4 | C22×D3 |
| Fitting | C2×C3×D8 | C2 |
| socle | C22×C3 | C2×D4 |
| hypercenter | C2×D8 | D3 |
| hypocenter | C3 | #459 |
| LCS | G ▷ C2×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C2×C4 ◁ C2×D8 | |
140 conjugacy classes of subgroups; only the 43 normal subgroups are shown.
This group has 140 conjugacy classes of subgroups — too many to draw. Only its 43 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 15 points:
(1,2,3)(5,7)(9,12)(10,15)(13,14);(8,9)(10,12)(11,14)(13,15);(2,3)(4,5,6,7)Polycyclic presentation:
2^6,3|a^2=e,d^2=f,c^a=ce,g^a=g^2,c^b=cd,d^b=df,d^c=df33 conjugacy classes — view the full character table.
| maximal subgroup of | #79567, #79569, #79570, … |
| maximal normal subgroup of | #79567, #79569, #79570, … |
| maximal quotient of | #70153, #70161, #70165, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 709);