Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | Aut(G) = #72260825 (order 4096) · Inn(G) = C22⋊C4 · Out(G) = C22×B(4,𝔽2) |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22⋊C4 |
| commutator | C22 | C22×C8 |
| Frattini, cosocle | C22×C4 | C23 |
| socle | C22 | C2×C22⋊C4 |
| prime radical | C22×D4 | C4 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
156 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2,3,6,4,5,7,8)(9,10,12,15,13,16,11,14);(2,5)(3,7);(9,11,13,12)(10,14,16,15)Polycyclic presentation:
2^7|a^2=e,c^2=fg,e^2=g,b^a=bd,d^a=df,e^b=ef44 conjugacy classes — view the full character table.
| maximal subgroup of | #7204, #7285, C8×#100, … |
| maximal normal subgroup of | #7204, #7285, C8×#100, … |
| maximal quotient of | #7567, #7569, #7571, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 192);