Finite Group DatabasebetaFinite Group Databasebeta| order | 64 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | Aut(G) = #120×#1020 (order 3072) · Inn(G) = C24 · Out(G) = D4×S4 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C22 | C24 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
100 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 16 points:
(2,5)(6,7)(9,10,12,14)(11,15,16,13);(1,2,4,5)(3,7,8,6);(9,11,12,16)(10,13,14,15);(1,3,4,8)(2,6,5,7)Polycyclic presentation:
2^6|a^2=f,b^2=e,c^2=f,d^2=e,b^a=be,c^a=cf,d^b=de25 conjugacy classes — view the full character table.
| Fitting subgroup of | #557⋊C3, #73093, #73094, … |
| maximal subgroup of | #1626, #1636, #1648, … |
| maximal normal subgroup of | #1626, #1636, #1648, … |
| maximal quotient of | Q8×C4⋊C4, #2351, #2355, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(64, 238);