Finite Group DatabasebetaFinite Group Databasebeta| order | 2400 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C52 ⋊ GL(2,ℤ/4) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D5×C5⋊S4 |
| commutator, cosocle | C2×C52×A4 | C22 |
| Fitting, socle | C23×C52 | C2×D3 |
| hypocenter | C52×A4 | D4 |
| LCS | G ▷ C2×C52×A4 ▷ C52×A4 | |
| UCS | C1 ◁ C2 | |
286 conjugacy classes of subgroups; only the 34 normal subgroups are shown.
This group has 286 conjugacy classes of subgroups — too many to draw. Only its 34 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 120 |
A minimal faithful permutation representation — 2 generators on 18 points:
(1,2,3,4)(5,7,8,6)(10,11)(12,13)(14,15)(16,17);(2,4)(7,8)(9,10,12,13,11)(15,16)(17,18)Polycyclic presentation:
2^3,3,5^2,2^2|b^2=c,b^a=bc,d^a=d^2,d^b=d^2,e^a=e^4,e^b=e^4,f^a=e^3f,f^b=f^4,g^d=h,h^a=gh,h^b=gh,h^d=gh98 conjugacy classes — view the full character table.
| maximal subgroup of | C22⋊D5×C5⋊S4, #72260214, #72260244, … |
| maximal normal subgroup of | C22⋊D5×C5⋊S4, #72260214, #72260244, … |
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");;
gap> coll := SingleCollector(F, [2, 2, 2, 3, 5, 5, 2, 2]);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 4, 2, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.5^4);;
gap> SetConjugate(coll, 5, 2, F.5^4);;
gap> SetConjugate(coll, 6, 1, F.5^3*F.6);;
gap> SetConjugate(coll, 6, 2, F.6^4);;
gap> SetConjugate(coll, 7, 4, F.8);;
gap> SetConjugate(coll, 8, 1, F.7*F.8);;
gap> SetConjugate(coll, 8, 2, F.7*F.8);;
gap> SetConjugate(coll, 8, 4, F.7*F.8);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);