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#72229366

order5760
automorphism?
homologyAbelianization (H1) = C4 · Schur multiplier (H2) = C2×C3

Aliases

No aliases for this group.

Splittings

FittingC5×H(5,𝔽2) ⋊ C3⋊Dic3
derivedC5×Ω+(4,𝔽3) ⋊ C4
SylowC5 ⋊ #11921471

Subgroups

subgroupG/H
center, Frattini, hypercenterC2#72224624
commutator, hypocenterC5×Ω+(4,𝔽3)C4
FittingC5×H(5,𝔽2)C3⋊Dic3
socleC2×C5A4⋊SL(2,ℤ/4)
prime radical, cosocleD5×Ω+(4,𝔽3)C2
LCSG ▷ C5×Ω+(4,𝔽3)
UCSC1 ◁ C2

502 conjugacy classes of subgroups; only the 38 normal subgroups are shown.

normal subgroups only

This group has 502 conjugacy classes of subgroups — too many to draw. Only its 38 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) = 29
transitive degreeμt(G) = 120

A minimal faithful permutation representation — 3 generators on 29 points:

(1,3,6,11)(2,24,13,7)(4,14)(5,12)(8,17,23,15)(9,18,21,22)(10,16,20,19)(26,27,28,29);(1,2,7)(3,9,22)(4,13,15)(5,8,16)(6,10,19)(11,23,17)(12,21,24)(14,20,18)(25,26,29,27,28);(3,4,5)(8,9,13)(11,14,12)(15,16,22)(17,18,24)(20,21,23)

Polycyclic presentation:

2^2,3^2,5,2^5|a^2=b,f^2=j,g^2=j,c^a=c^2,d^a=d^2,e^a=e^3,e^b=e^4,f^a=fj,f^c=g,f^d=g,g^a=fg,g^c=fgj,g^d=fgj,g^f=gj,h^a=ghij,h^c=i,h^d=fgh,h^g=hj,i^a=fij,i^c=hi,i^d=fij,i^f=ij

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽5)×SO+(4,𝔽3), C2×#72229366
maximal normal subgroup ofAGL(1,𝔽5)×SO+(4,𝔽3), C2×#72229366
maximal quotient ofC2×#72229366

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