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 | C23 | C22×C4 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
124 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,4,7,5,8,3,6)(9,10)(11,17,19,14)(12,15)(13,16)(18,23,24,21)(20,22);(9,11,13,19)(10,14,16,17)(12,18,20,24)(15,21,22,23);(1,3,5,4)(2,6,8,7)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)Polycyclic presentation:
2^7|a^2=e,b^2=f,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 | #7206, #7286, C8×#101, … |
| maximal normal subgroup of | #7206, #7286, C8×#101, … |
| maximal quotient of | #7568, #7570, #7585, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 193);