#72229157

order	5832
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂×C₃

Aliases

No aliases for this group.

Splittings

Sylow

H(3,𝔽₃)² ⋊ D₄

Subgroups

	subgroup	G/H
center, hypercenter	C₂	#72214385
commutator, Fitting, cosocle	C₂×H(3,𝔽₃)²	C₂²
Frattini, socle	C₂×C₃²	C₃⋊#1113
hypocenter	H(3,𝔽₃)²	D₄
LCS	G ▷ C₂×H(3,𝔽₃)² ▷ H(3,𝔽₃)²
UCS	C₁ ◁ C₂

2044 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 2044 conjugacy classes of subgroups — too many to draw. Only its 93 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) = 22
transitive degree	μ_t(G) = 108

A minimal faithful permutation representation — 3 generators on 22 points:

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

Polycyclic presentation:

2^3,3^6|a^2=c,b^a=bc,d^a=d^2,e^b=e^2,e^d=eh,f^a=e^2f^2,f^b=f^2,f^d=fi,g^a=g^2,g^e=gh^2,g^f=gh^2,h^a=h^2,h^b=h^2,i^a=hi,i^b=i^2

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

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