Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #560 |
| automorphism | Aut(G) order 24576 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C23 | C24 |
| commutator | C22 | C23×C4 |
| prime radical | C2×#120 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
344 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,3,4)(6,9)(7,11)(13,14)(15,19)(16,17)(18,20);(6,9)(10,12)(14,17)(15,20);(5,6)(7,10)(8,9)(11,12)(13,14)(15,18)(16,17)(19,20);(5,7)(6,10)(8,11)(9,12)(13,15)(14,18)(16,20)(17,19)Polycyclic presentation:
2^7|a^2=g,b^a=be,c^a=cf,d^a=def,c^b=cef,d^b=de38 conjugacy classes — view the full character table.
| Fitting subgroup of | #2375⋊C3, #4625⋊C4, (#1020⋊C2)⋊C4 |
| maximal subgroup of | #12960, #13043, C2×#2375, … |
| maximal normal subgroup of | #12960, #13043, C2×#2375, … |
| maximal quotient of | #13827, #13831, #13832, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1107);