Finite Group DatabasebetaFinite Group Databasebeta| order | 64 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #465 |
| automorphism | Aut(G) = #72217072 (order 4096) · Inn(G) = C2×D4 · Out(G) = #23900 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | 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 | |
81 conjugacy classes of subgroups.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 12 points:
(2,4)(6,9)(7,10)(11,12);(5,6,10,12,8,11,7,9);(1,2,3,4)Polycyclic presentation:
2^6|b^2=df,c^2=e,d^2=f,b^a=bd,c^a=ce,d^a=df22 conjugacy classes — view the full character table.
| derived subgroup of | #94819, #94820, #153613, … |
| Frattini subgroup of | #7515, #7516, #7529, … |
| maximal subgroup of | #1357, #1361, #1408, … |
| maximal normal subgroup of | #1357, #1361, #1408, … |
| maximal quotient of | #1706, #1709, #1712, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(64, 174);