Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | Aut(G) = #72237495 (order 4096) · Inn(G) = C22⋊C4 · Out(G) = C22×B(4,𝔽2) |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
| direct product | C4×#100 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22⋊C4 |
| commutator | C22 | C2×C42 |
| Frattini, cosocle | C22×C4 | C23 |
| socle | C22 | C2×C22⋊C4 |
| prime radical | C22×D4 | C22 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
180 conjugacy classes of subgroups.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 12 points:
(1,2,3,4);(5,6,7,10,8,9,11,12);(6,9)(7,11)44 conjugacy classes — view the full character table.
| maximal subgroup of | #7203, #7282, C8×#100, … |
| maximal normal subgroup of | #7203, #7282, C8×#100, … |
| maximal quotient of | C4×#323, #8152, #8155, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 487);