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#72227353

order25992
automorphism?
homologyAbelianization (H1) = C22×C9 · Schur multiplier (H2) = C2

Aliases

No aliases for this group.

Splittings

Hall#72227351 ⋊ C9
Hall#15692433 ⋊ D4
SylowC192 ⋊ D4×C9

Subgroups

subgroupG/H
center, Frattini, hypercenterC2#20266599⋊C2
commutator, Fitting, socleC2×C192C22×C9
prime radical#72227352C3
cosocleC2×#11762245C22×C3
hypocenterC192D4×C9
LCSG ▷ C2×C192 ▷ C192
UCSC1 ◁ C2

231 conjugacy classes of subgroups; only the 26 normal subgroups are shown.

normal subgroups only

This group has 231 conjugacy classes of subgroups — too many to draw. Only its 26 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) = 42
transitive degreeμt(G) = 76

A minimal faithful permutation representation — 2 generators on 42 points:

(2,4)(6,17,18,11,20,16,14,10,19,7,13,23,9,21,12,15,8,22)(24,25,31,42,29,33,37,35,34,26,27,41,40,32,28,36,30,39);(1,2,3,4)(5,6,10,17,12,20,19,14,11,18,23,9,15,22,21,16,13,8,7)(25,26)(27,29)(28,30)(31,36)(32,37)(33,34)(35,38)(39,42)(40,41)

Polycyclic presentation:

2^2,3,2,3,19^2|a^2=d,c^3=e,b^a=bd,f^a=f^18,f^b=f^18,f^c=f^17,f^e=f^11,g^a=f^2g,g^b=g^18,g^c=g^17,g^e=g^11

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72227356
maximal normal subgroup of#72227356

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");;
gap> coll := SingleCollector(F, [2, 2, 3, 2, 3, 19, 19]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 3, F.5);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 6, 1, F.6^18);;
gap> SetConjugate(coll, 6, 2, F.6^18);;
gap> SetConjugate(coll, 6, 3, F.6^17);;
gap> SetConjugate(coll, 6, 5, F.6^11);;
gap> SetConjugate(coll, 7, 1, F.6^2*F.7);;
gap> SetConjugate(coll, 7, 2, F.7^18);;
gap> SetConjugate(coll, 7, 3, F.7^17);;
gap> SetConjugate(coll, 7, 5, F.7^11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);