Finite Group DatabasebetaFinite Group Databasebeta| order | 64 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | Aut(G) = C2×C4×S4 (order 192) · Inn(G) = C22 · Out(G) = C2×C4×D3 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C16 | C22 |
| commutator, socle | C2 | C22×C8 |
| Frattini, cosocle | C8 | C23 |
| prime radical | #41 | C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C16 ◁ G | |
| LECS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
42 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,4,8,5,9,14,20,6,10,15,21,16,22,26,30)(3,7,12,18,13,19,25,29,11,17,23,27,24,28,31,32);(1,3)(2,7)(4,12)(5,13)(6,11)(8,18)(9,19)(10,17)(14,25)(15,23)(16,24)(20,29)(21,27)(22,28)(26,31)(30,32);(3,11)(7,17)(12,23)(13,24)(18,27)(19,28)(25,31)(29,32)Polycyclic presentation:
2^6|a^2=d,d^2=e,e^2=f,c^b=cf40 conjugacy classes — view the full character table.
| derived subgroup of | #33982⋊C3, #33983⋊C3, #33984⋊C3 |
| Fitting subgroup of | #504⋊C3, #68006, #73038, … |
| maximal subgroup of | #1401, #2170, #2171, … |
| maximal normal subgroup of | #1401, #2170, #2171, … |
| maximal quotient of | #2108, #2113, #2151, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(64, 185);