#72253808

order	8748
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂×C₃

Aliases

No aliases for this group.

Splittings

Fitting, Sylow	#14552186 ⋊ C₂²
derived	#10624555 ⋊ C₂²×C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	#10624555	C₂²×C₃
Frattini, socle	C₃³	C₃×D₃×C₃⋊D₃
Fitting	#14552186	C₂²
LCS	G ▷ #10624555
UCS	C₁

2792 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 2792 conjugacy classes of subgroups — too many to draw. Only its 82 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) = 27
transitive degree	μ_t(G) = 162

A minimal faithful permutation representation — 3 generators on 27 points:

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

Polycyclic presentation:

2^2,3^7|f^3=g,d^a=d^2,e^a=e^2,e^b=e^2,e^d=ei,f^a=f^2g^2,f^c=fg,f^e=fh,g^a=g^2,h^b=h^2,i^b=i^2

Conjugacy classes & character table

111 conjugacy classes — view the full character table.

Relationships

maximal subgroup of

#72253827

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