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#72254584

order26112
automorphism?
homologyAbelianization (H1) = C16 · Schur multiplier (H2) = C22

Aliases

No aliases for this group.

Splittings

FittingC17×H(5,𝔽2) ⋊ C3⋊C16
derivedC17×#997 ⋊ C16
SylowC17 ⋊ #72254549

Subgroups

subgroupG/H
center, Frattini, hypercenterC2#72254555
commutator, hypocenterC17×#997C16
FittingC17×H(5,𝔽2)C3⋊C16
socleC2×C17#11714263
prime radicalD17×#997C8
cosocle#997×#3633C2
LCSG ▷ C17×#997
UCSC1 ◁ C2

830 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 830 conjugacy classes of subgroups — too many to draw. Only its 39 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) = 25
transitive degreeμt(G) = 136

A minimal faithful permutation representation — 2 generators on 25 points:

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

Polycyclic presentation:

2^4,3,17,2^5|a^2=b,b^2=c,c^2=d,g^2=k,h^2=k,e^a=e^2,f^a=f^12,f^b=f^8,f^c=f^13,f^d=f^16,g^a=gh,g^e=h,h^a=hk,h^e=ghk,h^g=hk,i^a=ij,i^e=j,i^h=ik,j^e=ij,j^g=jk

Conjugacy classes & character table

115 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#5935×AGL(1,𝔽17)
maximal normal subgroup of#5935×AGL(1,𝔽17)

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", "j", "k");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 3, 17, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.2);;
gap> SetPower(coll, 2, F.3);;
gap> SetPower(coll, 3, F.4);;
gap> SetPower(coll, 7, F.11);;
gap> SetPower(coll, 8, F.11);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.6^12);;
gap> SetConjugate(coll, 6, 2, F.6^8);;
gap> SetConjugate(coll, 6, 3, F.6^13);;
gap> SetConjugate(coll, 6, 4, F.6^16);;
gap> SetConjugate(coll, 7, 1, F.7*F.8);;
gap> SetConjugate(coll, 7, 5, F.8);;
gap> SetConjugate(coll, 8, 1, F.8*F.11);;
gap> SetConjugate(coll, 8, 5, F.7*F.8*F.11);;
gap> SetConjugate(coll, 8, 7, F.8*F.11);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 9, 5, F.10);;
gap> SetConjugate(coll, 9, 8, F.9*F.11);;
gap> SetConjugate(coll, 10, 5, F.9*F.10);;
gap> SetConjugate(coll, 10, 7, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);