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#72260363

order7200
automorphism?
homologyAbelianization (H1) = C2×C4 · Schur multiplier (H2) = C22

Aliases

No aliases for this group.

Splittings

Fitting, socleC22×C3×C52 ⋊ C4×D3
derivedC3×C52×A4 ⋊ C2×C4
SylowC52 ⋊ Dic3×S4

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, hypocenterC3×C52×A4C2×C4
Fitting, socleC22×C3×C52C4×D3
prime radicalC3×(C5⋊S4)⋊D5C2
cosocleC3×A4×C5⋊D5C22
LCSG ▷ C3×C52×A4
UCSC1

572 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 45 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) = 120

A minimal faithful permutation representation — 2 generators on 17 points:

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

Polycyclic presentation:

2^3,3^2,5^2,2^2|b^2=c,d^a=d^2,d^b=d^2,e^a=d^2e,e^b=e^2,f^a=f^4,f^b=f^3,f^c=f^4,g^a=f^2g,g^b=g^3,g^c=g^4,h^a=i,h^b=i,h^d=hi,h^e=i,i^a=h,i^b=h,i^d=h,i^e=hi

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72260373
maximal quotient of#72260373

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