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#72226420

order3200
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C24

Aliases

No aliases for this group.

Splittings

SylowC52 ⋊ #1906

Subgroups

subgroupG/H
centerC2#13104114
commutatorC2×C4×C52C22×C4
FrattiniC2×C4C2×D5⋊AGL(1,𝔽5)
FittingC52×#41C2×C4
socleC2×C52#386
prime radical#13106562C2
cosocleC2×C4×C5⋊D5C23
hypercenter#41D5⋊AGL(1,𝔽5)
hypocenterC52#1906
LCSG ▷ C2×C4×C52 ▷ C2×C52 ▷ C52
UCSC1 ◁ C2 ◁ C2×C4 ◁ #41

974 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 974 conjugacy classes of subgroups — too many to draw. Only its 87 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)(2,7)(4,12)(5,11)(6,9)(8,15)(10,13)(14,16)(17,18,21,19,20)(22,23,26,25,24);(1,3)(2,6,8,14)(4,12)(5,11)(7,9,15,16)(10,13)(18,19,20,21)(23,25,24,26);(1,2,4,9,5,8,13,16)(3,6,10,7,11,14,12,15)(23,24)(25,26)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72226421
maximal normal subgroup of#72226421

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