Finite Group DatabasebetaFinite Group Databasebeta| order | 64 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | Aut(G) = #72217947 (order 2048) · Inn(G) = C22 · Out(G) = C2×#23900 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
| direct product | C4×C4⋊C4 |
| unambiguous semidirect product | C4⋊C42 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22 |
| commutator | C2 | C2×C42 |
| Frattini, socle, prime radical, cosocle | C23 | C23 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
97 conjugacy classes of subgroups.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 12 points:
(1,2,3,4);(6,8)(9,10,11,12);(5,6,7,8)40 conjugacy classes — view the full character table.
| maximal subgroup of | #1279, #1285, #1303, … |
| maximal normal subgroup of | #1279, #1285, #1303, … |
| maximal quotient of | C4×#95, #1434, C4⋊#95, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(64, 59);