Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2204 |
| automorphism | Aut(G) = #72209660 (order 1536) · Inn(G) = B(4,𝔽2) · Out(G) = S4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×Q8 | C23 |
| prime radical | 2−1+4 | C22 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ G | |
| LECS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
112 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)(3,11)(4,14)(5,9)(6,16)(7,12)(8,13)(10,15);(1,3,7,6)(2,8,12,14)(4,15,13,5)(9,11,10,16);(1,4,7,13)(2,9,12,10)(3,5,6,15)(8,16,14,11)Polycyclic presentation:
2^7|b^2=g,c^2=g,d^2=g,e^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eg,f^a=fg,c^b=cg,d^b=dg,e^b=efg,d^c=df,e^c=eg,e^d=eg20 conjugacy classes — view the full character table.
| Fitting subgroup of | #2205⋊C3, #11711673, #11711674 |
| maximal subgroup of | #13683, #13684, C2×#2205, … |
| maximal normal subgroup of | #13683, #13684, C2×#2205, … |
| maximal quotient of | #12101, #12130, #12146, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 937);