Finite Group DatabasebetaFinite Group Databasebeta| order | 64 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | Aut(G) = #120×#1020 (order 3072) · Inn(G) = C24 · Out(G) = D4×S4 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C22 | C24 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
156 conjugacy classes of subgroups.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 12 points:
(2,4)(6,7)(10,11);(5,6,9,7)(8,11,12,10);(1,2)(3,4)(5,7,9,6)(8,11,12,10);(5,8,9,12)(6,10,7,11)Polycyclic presentation:
2^6|b^2=e,c^2=e,d^2=e,b^a=be,c^a=cf,d^b=de25 conjugacy classes — view the full character table.
| Fitting subgroup of | C4⋊(#41⋊C3), #73089, #73090, … |
| maximal subgroup of | #1621, #1631, #1641, … |
| maximal normal subgroup of | #1621, #1631, #1641, … |
| maximal quotient of | #2341, Q8×C4⋊C4, #2354, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(64, 231);