Finite Group DatabasebetaFinite Group Databasebeta| order | 312 |
| isoclinism | isoclinic class of Dic78 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting | C22×C3×C13 ⋊ C2 |
| Hall | C3×C13 ⋊ D4 |
| Sylow | C13 ⋊ #67 |
| Sylow | C3 ⋊ C22⋊D13 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×D39 |
| commutator, socle, cosocle | C2×C3×C13 | C22 |
| Fitting | C22×C3×C13 | C2 |
| hypercenter | C22 | D39 |
| hypocenter | C3×C13 | D4 |
| LCS | G ▷ C2×C3×C13 ▷ C3×C13 | |
| UCS | C1 ◁ C2 ◁ C22 | |
32 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 156 |
A minimal faithful permutation representation — 2 generators on 20 points:
(1,2,4,6,8,10,12,13,11,9,7,5,3)(14,15,16)(18,20);(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(15,16)(17,18)(19,20)Polycyclic presentation:
2^3,3,13|b^a=bc,d^a=d^2,e^a=e^1281 conjugacy classes — view the full character table.
| maximal subgroup of | #10597745, #10597748, #67×D13, … |
| maximal normal subgroup of | #10597745, #10597748, #67×D13, … |
| maximal quotient of | #10597664, #10597666, #10597667, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(312, 41);