#72243995

order	8748
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂×C₃²

Aliases

No aliases for this group.

Splittings

Fitting, Sylow	C₃×#10624930 ⋊ C₂²
derived	C₃²×AΣL(1,𝔽₂₇) ⋊ C₂²×C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₃²×AΣL(1,𝔽₂₇)	C₂²×C₃
Frattini	C₃²	C₃×C₃⋊D₃²
Fitting	C₃×#10624930	C₂²
socle	C₃²	C₃×((C₃⋊D₃)⋊C₃)⋊D₃
LCS	G ▷ C₃²×AΣL(1,𝔽₂₇)
UCS	C₁

3770 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 3770 conjugacy classes of subgroups — too many to draw. Only its 110 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) = 30
transitive degree	μ_t(G) = 54

A minimal faithful permutation representation — 3 generators on 30 points:

(1,2)(4,13,12)(5,26,22,9,21,20)(6,23,8)(7,14,24)(10,17,28,16,11,29)(15,25,30,18,19,27);(2,3)(5,9)(6,12)(7,8)(10,18)(11,20)(13,23)(14,24)(15,16)(17,22)(19,27)(21,29)(25,30)(26,28);(4,10,18,7,5,16,8,15,9)(6,19,26,13,11,25,14,21,17)(12,28,27,24,22,29,23,30,20)

Polycyclic presentation:

2^2,3^7|d^a=d^2,e^a=e^2,e^b=e^2,e^d=eh,f^a=f^2,f^c=fi^2,g^a=g^2,g^b=g^2i,h^a=hi^2,h^b=h^2i,h^e=hi,i^a=i^2

Conjugacy classes & character table

135 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72244009, D₃×#72221953
maximal normal subgroup of	D₃×#72221953

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