Finite Group DatabasebetaFinite Group Databasebeta| order | 4032 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| derived | C7×C22⋊A4 ⋊ C3×C4 |
| Sylow | C7 ⋊ C3×#5936 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | #72214032 |
| commutator, hypocenter | C7×C22⋊A4 | C3×C4 |
| Fitting, socle | C25×C7 | C3×D3 |
| prime radical | C2×ΓL(1,𝔽8)×C22⋊A4 | C2 |
| cosocle | C2×C7×C22⋊A4 | C2×C3 |
| LCS | G ▷ C7×C22⋊A4 | |
| UCS | C1 ◁ C2 | |
628 conjugacy classes of subgroups; only the 38 normal subgroups are shown.
This group has 628 conjugacy classes of subgroups — too many to draw. Only its 38 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 84 |
A minimal faithful permutation representation — 2 generators on 19 points:
(1,3,4,2)(5,8,7,6)(9,10,11,12)(14,18,17,15,16,19);(2,3,4)(6,7,8)(13,14,17,18,16,19,15)Polycyclic presentation:
2,3,2,3,7,2^4|a^2=c,d^a=d^2,e^a=e^6,e^b=e^4,f^a=g,f^d=g,g^a=f,g^d=fg,h^a=hi,h^d=i,i^d=hi76 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽7)×#5936, #72276650, #72276910, … |
| maximal normal subgroup of | AGL(1,𝔽7)×#5936, #72276650, #72276910, … |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:
gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i");;
gap> coll := SingleCollector(F, [2, 3, 2, 3, 7, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.3);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.5^6);;
gap> SetConjugate(coll, 5, 2, F.5^4);;
gap> SetConjugate(coll, 6, 1, F.7);;
gap> SetConjugate(coll, 6, 4, F.7);;
gap> SetConjugate(coll, 7, 1, F.6);;
gap> SetConjugate(coll, 7, 4, F.6*F.7);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 4, F.9);;
gap> SetConjugate(coll, 9, 4, F.8*F.9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);