#72230580

order	31752
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Fitting, socle, derived	C₃³×C₇² ⋊ C₂³×C₃
Hall	C₃³×#3861 ⋊ C₂³
Sylow	C₇² ⋊ C₃×D₃³

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃³×C₇²	C₂³×C₃
LCS	G ▷ C₃³×C₇²
UCS	C₁

2364 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 2364 conjugacy classes of subgroups — too many to draw. Only its 117 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 84

A minimal faithful permutation representation — 3 generators on 23 points:

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

Polycyclic presentation:

2^3,3^4,7^2|e^a=e^2,e^c=e^2,f^a=e^2f,f^c=f^2,g^b=g^2,h^a=h^6,h^b=h^6,h^c=h^6,h^d=h^2,i^a=h^2i,i^b=i^6,i^c=i^6,i^d=i^2

Conjugacy classes & character table

162 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#72213636
maximal normal subgroup of	D₃×#72213636

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