#72227929

order	2916
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₃⁴

Aliases

No aliases for this group.

Splittings

Fitting, derived, Sylow

H(3,𝔽₉) ⋊ C₄

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, hypocenter	H(3,𝔽₉)	C₄
Frattini, socle	C₃²	C₃⋊(C₃⋊#157)
prime radical, cosocle	#13091538	C₂
LCS	G ▷ H(3,𝔽₉)
UCS	C₁

307 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 54
transitive degree	μ_t(G) = 54

A minimal faithful permutation representation — 3 generators on 54 points:

(1,2)(3,8,12,27)(4,16,17,45)(5,22,23,38)(6,11,7,26)(9,31)(10,35,36,25)(13,42,24,33)(14,30,41,47)(15,44,40,52)(18,46)(19,48,39,34)(20,28,43,29)(21,37,51,54)(32,50,53,49);(1,23,35)(2,36,45)(3,4,49)(5,31,15)(6,39,43)(7,17,51)(8,32,22)(9,48,44)(10,33,52)(11,29,38)(12,24,20)(13,21,41)(14,19,50)(16,30,53)(18,25,40)(26,54,34)(27,28,42)(37,46,47);(1,18,31)(2,53,42)(3,23,20)(4,51,40)(5,43,12)(6,13,49)(7,24,50)(8,37,34)(9,33,11)(10,38,44)(14,39,25)(15,17,21)(16,27,36)(19,35,41)(22,47,54)(26,32,46)(28,45,30)(29,48,52)

Polycyclic presentation:

2^2,3^6|a^2=b,c^a=cd,c^b=c^2,d^a=cd^2,d^b=d^2,e^a=c^2d^2e^2h^2,e^b=c^2degh,e^c=eg,e^d=egh^2,f^a=cd^2f^2g,f^b=c^2fg,f^c=fg^2h^2,f^d=fh,f^e=fg^2h^2,g^a=gh,g^b=g^2,h^a=gh^2,h^b=h^2

Conjugacy classes & character table

32 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72227930
maximal normal subgroup of	#72227930

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