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#72226549

order3200
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C25×C5

Aliases

No aliases for this group.

Splittings

FittingC22×C52×D4 ⋊ C4
SylowC52 ⋊ #1795

Subgroups

subgroupG/H
centerC2#13104938
commutatorC2×C4×C52C22×C4
FrattiniC2×C4C22×#1044
FittingC22×C52×D4C4
socleC2×C52#379
prime radicalC2×#11717517C2
cosocleC2×C4×C5⋊D5C23
hypercenterC22×D4#1044
hypocenterC52#1795
LCSG ▷ C2×C4×C52 ▷ C2×C52 ▷ C52
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×D4

2074 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 2074 conjugacy classes of subgroups — too many to draw. Only its 114 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 80

A minimal faithful permutation representation — 3 generators on 26 points:

(1,3)(2,8)(4,12)(5,11)(6,13)(7,14)(9,15)(10,16)(17,18,21,19,20)(22,23,24,26,25);(2,7)(4,5)(8,14)(9,10)(11,12)(15,16);(1,2,4,9,6,10,5,7)(3,8,11,14,13,16,12,15)(18,19,20,21)(23,24,25,26)

Polycyclic presentation:

2^7,5^2|a^2=d,d^2=g,e^2=g,f^2=g,b^a=be,c^a=cefg,d^b=dg,d^c=dg,e^b=eg,f^a=fg,f^b=fg,h^a=h^2,h^d=h^4,i^a=h^3i^3,i^d=i^4

Conjugacy classes & character table

149 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72226551
maximal normal subgroup of#72226551

GAP

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, 2, 2, 2, 2, 2, 5, 5]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 5, F.7);;
gap> SetPower(coll, 6, F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.5*F.6*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.7);;
gap> SetConjugate(coll, 6, 1, F.6*F.7);;
gap> SetConjugate(coll, 6, 2, F.6*F.7);;
gap> SetConjugate(coll, 8, 1, F.8^2);;
gap> SetConjugate(coll, 8, 4, F.8^4);;
gap> SetConjugate(coll, 9, 1, F.8^3*F.9^3);;
gap> SetConjugate(coll, 9, 4, F.9^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);