Finite Group Databasebeta

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#72243964

order7680
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C25

Aliases

No aliases for this group.

Splittings

FittingC5×#379 ⋊ C4×D3
SylowC5 ⋊ #72243948

Subgroups

subgroupG/H
center, hypercenterC2AGL(1,𝔽5)×#1020⋊C2
commutatorC2×C5×C22⋊A4C22×C4
FrattiniC23C2×AGL(1,𝔽5)×S4
FittingC5×#379C4×D3
socleC23×C5C2×C4×S4
prime radicalD5×#72991C2
cosocleC2×D5×C22⋊A4C23
hypocenterC5×C22⋊A4C4×D4
LCSG ▷ C2×C5×C22⋊A4 ▷ C5×C22⋊A4
UCSC1 ◁ C2

5274 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 5274 conjugacy classes of subgroups — too many to draw. Only its 74 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) = 17
transitive degreeμt(G) = 60

A minimal faithful permutation representation — 3 generators on 17 points:

(1,2,4,3)(7,9)(11,17,15)(12,13,16);(2,3,4,5)(12,16)(14,17);(2,3,4,5)(6,7)(8,9)(10,11)(12,14)(13,15)(16,17)

Polycyclic presentation:

2^5,3,5,2^4|a^2=d,b^2=d,c^2=d,c^a=ce,f^b=f^2,g^a=g^3,g^b=g^3,g^c=g^3,g^d=g^4,h^a=hj,h^b=hi,h^f=i,i^a=ik,i^f=hi,j^b=jk,j^f=k,k^f=jk

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽5)×#11713200
maximal normal subgroup ofAGL(1,𝔽5)×#11713200

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, 2, 3, 5, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 2, F.4);;
gap> SetPower(coll, 3, F.4);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 6, 2, F.6^2);;
gap> SetConjugate(coll, 7, 1, F.7^3);;
gap> SetConjugate(coll, 7, 2, F.7^3);;
gap> SetConjugate(coll, 7, 3, F.7^3);;
gap> SetConjugate(coll, 7, 4, F.7^4);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 8, 6, F.9);;
gap> SetConjugate(coll, 9, 1, F.9*F.11);;
gap> SetConjugate(coll, 9, 6, F.8*F.9);;
gap> SetConjugate(coll, 10, 2, F.10*F.11);;
gap> SetConjugate(coll, 10, 6, F.11);;
gap> SetConjugate(coll, 11, 6, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);