Finite Group DatabasebetaFinite Group Databasebeta| order | 2688 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C2×C7×H(5,𝔽2) ⋊ C2×C3 |
| Hall | #11744560 ⋊ C3 |
| Sylow | C7 ⋊ #87547 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×D7×C22⋊A4 |
| commutator, cosocle, hypocenter | C7×H(5,𝔽2) | C22×C3 |
| Fitting | C2×C7×H(5,𝔽2) | C2×C3 |
| socle | C2×C7 | C22×C22⋊A4 |
| hypercenter | C22 | D7×C22⋊A4 |
| LCS | G ▷ C7×H(5,𝔽2) | |
| UCS | C1 ◁ C2 ◁ C22 | |
2154 conjugacy classes of subgroups; only the 58 normal subgroups are shown.
This group has 2154 conjugacy classes of subgroups — too many to draw. Only its 58 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 112 |
A minimal faithful permutation representation — 3 generators on 23 points:
(2,3)(4,5)(6,7)(9,13)(10,16,11)(14,23,15,20,22,21)(17,18,19);(1,2,4,6,7,5,3)(8,9)(10,20)(11,21)(12,13)(14,17)(15,19)(16,22)(18,23);(8,18,12,16)(9,23,13,22)(10,11,17,19)(14,15,20,21)Polycyclic presentation:
2^2,3,2^4,7,2|d^2=i,e^2=i,b^a=bi,d^c=e,e^c=dei,e^d=ei,f^c=fg,f^e=fi,g^c=f,g^d=gi,g^e=gi,h^b=h^6101 conjugacy classes — view the full character table.
| maximal subgroup of | #72259707 |
| maximal normal subgroup of | #72259707 |
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, 2, 3, 2, 2, 2, 2, 7, 2]);;
gap> SetPower(coll, 4, F.9);;
gap> SetPower(coll, 5, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.9);;
gap> SetConjugate(coll, 4, 3, F.5);;
gap> SetConjugate(coll, 5, 3, F.4*F.5*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.7);;
gap> SetConjugate(coll, 6, 5, F.6*F.9);;
gap> SetConjugate(coll, 7, 3, F.6);;
gap> SetConjugate(coll, 7, 4, F.7*F.9);;
gap> SetConjugate(coll, 7, 5, F.7*F.9);;
gap> SetConjugate(coll, 8, 2, F.8^6);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);