Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting, Sylow | C22×#3026 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×#85645 |
| commutator, cosocle, hypocenter | C22×H(5,𝔽2) | C22×C3 |
| Frattini | C23 | C22×C22⋊A4 |
| Fitting | C22×#3026 | C3 |
| socle | C23 | C2×#120⋊C3 |
| hypercenter | C22 | #85645 |
| LCS | G ▷ C22×H(5,𝔽2) | |
| UCS | C1 ◁ C2 ◁ C22 | |
8318 conjugacy classes of subgroups; only the 52 normal subgroups are shown.
This group has 8318 conjugacy classes of subgroups — too many to draw. Only its 52 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 48 |
Polycyclic presentation:
2^2,3,2^7|b^a=bj,d^a=dh,d^c=e,e^a=ei,e^c=de,f^c=g,g^c=fg,h^c=i,h^e=hj,i^c=hij,i^d=ij52 conjugacy classes — view the full character table.
| maximal subgroup of | #72258360, #72258614, #72258646, … |
| maximal normal subgroup of | #72258360, #72311942, #72312025 |
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", "j");;
gap> coll := SingleCollector(F, [2, 2, 3, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetConjugate(coll, 2, 1, F.2*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 4, 3, F.5);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> SetConjugate(coll, 5, 3, F.4*F.5);;
gap> SetConjugate(coll, 6, 3, F.7);;
gap> SetConjugate(coll, 7, 3, F.6*F.7);;
gap> SetConjugate(coll, 8, 3, F.9);;
gap> SetConjugate(coll, 8, 5, F.8*F.10);;
gap> SetConjugate(coll, 9, 3, F.8*F.9*F.10);;
gap> SetConjugate(coll, 9, 4, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);