#72243929

order	2916
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂×C₃

Aliases

No aliases for this group.

Splittings

Fitting, Sylow	C₃×H(5,𝔽₃) ⋊ C₂²
derived	C₃²×H(3,𝔽₃) ⋊ C₂²×C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₃²×H(3,𝔽₃)	C₂²×C₃
Frattini	C₃	C₃×C₃⋊#1113
Fitting	C₃×H(5,𝔽₃)	C₂²
socle	C₃²	C₃×#1113
LCS	G ▷ C₃²×H(3,𝔽₃)
UCS	C₁

1681 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 1681 conjugacy classes of subgroups — too many to draw. Only its 74 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) = 81

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

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

Polycyclic presentation:

2^2,3^6|d^b=d^2,d^c=dh,e^a=e^2,f^a=fg,f^b=f^2,f^e=fh,g^a=g^2,g^b=g^2,g^e=gh,h^b=h^2

Conjugacy classes & character table

105 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72243931, #72243949, D₃×#66432⋊C₃, …
maximal normal subgroup of	#72243949, D₃×#66432⋊C₃, #72277838
maximal quotient of	#72277786

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