Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | Aut(G) order 6144 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C4 | C24 |
| commutator, socle | C22 | C23×C4 |
| prime radical | C2×#41 | C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
201 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 16 points:
(1,2)(3,6)(4,5)(7,8)(9,10,14,11,13,15,12,16);(2,5)(6,8)(10,15)(11,16);(9,11,14,15,13,16,12,10);(1,3,4,7)(2,6,5,8)(9,12,13,14)(10,11,15,16)Polycyclic presentation:
2^7|a^2=f,c^2=f,d^2=e,f^2=g,b^a=be,c^a=cg,d^a=dg,c^b=cg,d^c=dg50 conjugacy classes — view the full character table.
| Fitting subgroup of | #2971⋊C3, #11711166, #11711167, … |
| maximal subgroup of | #9160, #9162, #9350, … |
| maximal normal subgroup of | #9160, #9162, #9350, … |
| maximal quotient of | #20480, #20494, Q8×#97, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1703);