Finite Group DatabasebetaFinite Group Databasebeta| order | 64 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #560 |
| automorphism | Aut(G) = #72226554 (order 9216) · Inn(G) = C24 · Out(G) = #10593244 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
| unitriangular | UT(3,𝔽4) |
| Heisenberg | H(3,𝔽4) |
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C22 | C24 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
130 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 4 generators on 16 points:
(1,2,5,8)(3,10,11,7)(4,12,14,16)(6,9,15,13);(1,3,5,11)(2,7,8,10)(4,13,14,9)(6,12,15,16);(1,4)(2,3)(5,14)(6,7)(8,11)(9,12)(10,15)(13,16);(1,3)(2,4)(5,11)(6,12)(7,9)(8,14)(10,13)(15,16)Polycyclic presentation:
2^6|a^2=e,b^2=e,b^a=be,c^a=cf,d^a=def,c^b=cef,d^b=de19 conjugacy classes — view the full character table.
| derived subgroup of | #5464, #5465, #1655⋊C3, … |
| Fitting subgroup of | #5450, #5464, #5465, … |
| maximal subgroup of | #1655, #1659, #2199, … |
| maximal normal subgroup of | #1655, #1659, #2199, … |
| maximal quotient of | #2375, #2379, #2670, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(64, 242);