Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C22×#376 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×#85547 |
| commutator, cosocle, hypocenter | C22⋊(#95⋊C3) | C22 |
| Frattini | C23 | C2×#1020 |
| Fitting | C22×#376 | D3 |
| socle | C25 | C2×S4 |
| hypercenter | C4 | #85547 |
| LCS | G ▷ C22⋊(#95⋊C3) | |
| UCS | C1 ◁ C2 ◁ C4 | |
1344 conjugacy classes of subgroups; only the 26 normal subgroups are shown.
This group has 1344 conjugacy classes of subgroups — too many to draw. Only its 26 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 2 generators on 28 points:
(1,2,4,3)(5,9)(6,10)(7,26,16,23)(11,28,19,25)(12,17)(13,27,24,15)(14,21,22,20);(2,3,4)(5,11,24,9,7,21,10,19,13,6,16,20)(8,22,28,17,15,23,18,14,25,12,27,26)Polycyclic presentation:
2^2,3,2^7|a^2=j,d^2=h,e^2=i,b^a=bj,c^b=c^2,d^b=deij,d^c=ehj,e^b=ei,e^c=dej,e^d=ej,f^b=ghj,f^c=g,g^b=fhi,g^c=fg,h^b=hij,h^c=i,i^c=hij56 conjugacy classes — view the full character table.
| maximal subgroup of | #72239827 |
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> SetPower(coll, 1, F.10);;
gap> SetPower(coll, 4, F.8);;
gap> SetPower(coll, 5, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.10);;
gap> SetConjugate(coll, 3, 2, F.3^2);;
gap> SetConjugate(coll, 4, 2, F.4*F.5*F.9*F.10);;
gap> SetConjugate(coll, 4, 3, F.5*F.8*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 5, 3, F.4*F.5*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 2, F.7*F.8*F.10);;
gap> SetConjugate(coll, 6, 3, F.7);;
gap> SetConjugate(coll, 7, 2, F.6*F.8*F.9);;
gap> SetConjugate(coll, 7, 3, F.6*F.7);;
gap> SetConjugate(coll, 8, 2, F.8*F.9*F.10);;
gap> SetConjugate(coll, 8, 3, F.9);;
gap> SetConjugate(coll, 9, 3, F.8*F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);