Finite Group DatabasebetaFinite Group Databasebeta| order | 64 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | Aut(G) = #72214271 (order 1024) · Inn(G) = D4 · Out(G) = #2403 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
| direct product | C2×#103 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | D4 |
| commutator | C4 | C22×C4 |
| Frattini, cosocle | C2×C4 | C23 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
81 conjugacy classes of subgroups.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 14 points:
(1,2);(3,4,5,6)(8,11)(9,13)(12,14);(7,8,10,12)(9,14,13,11)28 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#103, #1763, #1787, … |
| maximal normal subgroup of | C4×#103, #1763, #1787, … |
| maximal quotient of | C2×Q8⋊C8, #1481, #1487, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(64, 96);