Finite Group DatabasebetaFinite Group Databasebeta| order | 312 |
| isoclinism | isoclinic class of D3×D13 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| derived, Hall | C3×C13 ⋊ C2×C4 |
| Sylow | C13 ⋊ C4×D3 |
| Sylow | C3 ⋊ C4×D13 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D3×D13 |
| commutator, hypocenter | C3×C13 | C2×C4 |
| Fitting, socle, cosocle | C2×C3×C13 | C22 |
| prime radical | C2×D39 | C2 |
| LCS | G ▷ C3×C13 | |
| UCS | C1 ◁ C2 | |
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)(15,16)(17,19,20,18);(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15,16)(17,18,20,19)Polycyclic presentation:
2^3,3,13|a^2=c,b^2=c,e^a=e^12,d^b=d^248 conjugacy classes — view the full character table.
| maximal subgroup of | #10597691, #10597693, #10597735, … |
| maximal normal subgroup of | #10597691, #10597693, #10597735, … |
| maximal quotient of | #10597591, #10597594, Dic3×Dic13, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(312, 17);