Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2069 |
| automorphism | Aut(G) = #72237738 (order 3072) · Inn(G) = #396 · Out(G) = C22×A4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #396 |
| commutator, Frattini, cosocle | C22×C4 | C23 |
| prime radical | C22×D4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
108 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2,6,8,5,9,4,13)(3,10,14,15,11,16,12,7);(1,3,4,12,5,11,6,14)(2,7,8,10,9,15,13,16);(3,11)(4,6)(8,13)(10,16)Polycyclic presentation:
2^7|a^2=de,b^2=d,d^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eg,f^a=fg,c^b=cf,f^b=fg,d^c=dg23 conjugacy classes — view the full character table.
| Fitting subgroup of | #2069⋊C3 |
| maximal subgroup of | C2×#2069, #23622, #24808, … |
| maximal normal subgroup of | C2×#2069, #23622, #24808, … |
| maximal quotient of | #10603, #10605, #10607, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 801);