#72241662

order	7992
automorphism	?
homology	Abelianization (H₁) = C₂×C₉ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Fitting, socle	C₂²×C₃₇ ⋊ D₃×C₉
derived	A₄×C₃₇ ⋊ C₂×C₉
Sylow	C₃₇ ⋊ C₉×S₄

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, hypocenter	A₄×C₃₇	C₂×C₉
Fitting, socle	C₂²×C₃₇	D₃×C₉
prime radical	#72241650	C₃
cosocle	A₄×C₃₇⋊C₃	C₂×C₃
LCS	G ▷ A₄×C₃₇
UCS	C₁

74 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 41
transitive degree	μ_t(G) = 148

A minimal faithful permutation representation — 2 generators on 41 points:

(1,4,3,2)(6,22,36,12,26,21,15,11,24,7,14,34,9,28,13,16,8,20)(10,27,23,30,41,37,33,29,35,17,32,25,19,40,31,38,18,39);(2,3,4)(5,6,10,20,34,40,39,13,25,8,15,12,14,27,31,19,26,38,18,29,33,28,30,37,32,22,9,16,11,23,21,35,41,36,24,17,7)

Polycyclic presentation:

2,3^3,37,2^2|b^3=c,d^a=d^2,e^a=e^36,e^b=e^9,e^c=e^26,f^a=fg,f^d=g,g^d=fg

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	S₄×#66526⋊C₂
maximal normal subgroup of	S₄×#66526⋊C₂

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");;
gap> coll := SingleCollector(F, [2, 3, 3, 3, 37, 2, 2]);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.5^36);;
gap> SetConjugate(coll, 5, 2, F.5^9);;
gap> SetConjugate(coll, 5, 3, F.5^26);;
gap> SetConjugate(coll, 6, 1, F.6*F.7);;
gap> SetConjugate(coll, 6, 4, F.7);;
gap> SetConjugate(coll, 7, 4, F.6*F.7);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);