Finite Group DatabasebetaFinite Group Databasebeta| order | 64 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468 |
| automorphism | Aut(G) = #72213256 (order 1024) · Inn(G) = C2×D4 · Out(G) = B(4,𝔽2) |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
45 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)(3,6)(4,5)(7,8)(9,10,15,20)(11,17,22,23)(12,16,21,24)(13,19,14,18);(2,5)(6,8)(9,11,13,12,15,22,14,21)(10,16,18,17,20,24,19,23);(1,3,4,7)(2,6,5,8)(9,12,13,22,15,21,14,11)(10,17,18,24,20,23,19,16)Polycyclic presentation:
2^6|a^2=f,b^2=ef,c^2=d,d^2=f,e^2=f,b^a=bd,c^a=ce,d^a=df,e^a=ef,c^b=cf16 conjugacy classes — view the full character table.
| Frattini subgroup of | #7525, #7526, C2×#7525, … |
| maximal subgroup of | #1356, #1413, #1662, … |
| maximal normal subgroup of | #1356, #1413, #1662, … |
| maximal quotient of | #1836, #1935, #1960, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(64, 172);