#72228150

order	2916
automorphism	?
homology	Abelianization (H₁) = C₃×C₄ · Schur multiplier (H₂) = C₃³

Aliases

No aliases for this group.

Splittings

derived	H(5,𝔽₃) ⋊ C₃×C₄
Sylow	#10624761 ⋊ C₄

Subgroups

	subgroup	G/H
center, socle, hypercenter	C₂×C₃	#92301
commutator, hypocenter	H(5,𝔽₃)	C₃×C₄
Frattini	C₂×C₃³	C₃×C₃⋊D₃
Fitting, prime radical	C₂×#10624761	C₂
cosocle	C₂×H(5,𝔽₃)	C₂×C₃
LCS	G ▷ H(5,𝔽₃)
UCS	C₁ ◁ C₂×C₃

554 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 31
transitive degree	μ_t(G) = 108

A minimal faithful permutation representation — 3 generators on 31 points:

(2,13,9,6,7,19)(3,20,12,10,11,21)(8,26,15,16,17,24)(14,25,23,18,22,27)(28,29,30,31);(1,3,10)(2,8,18)(4,11,20)(5,12,21)(6,14,16)(7,15,25)(9,17,27)(13,22,24)(19,23,26);(1,2,6)(3,8,14)(4,9,19)(5,7,13)(10,18,16)(11,17,23)(12,15,22)(20,27,26)(21,25,24)

Polycyclic presentation:

2,3,2,3^5|a^2=c,d^a=d^2,d^b=df,e^a=e^2,e^b=eg,f^a=f^2h,f^d=fh,f^e=fh,g^a=g^2h^2,g^d=gh,g^e=gh^2

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72228151
maximal normal subgroup of	#72228151

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