#72260232

order	4800
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Sylow

C₅² ⋊ D₄×S₄

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₅²×S₄
commutator, cosocle	C₂×C₅²×A₄	C₂³
Fitting, socle	C₂³×C₅²	C₂²×D₃
hypocenter	C₅²×A₄	C₂×D₄
LCS	G ▷ C₂×C₅²×A₄ ▷ C₅²×A₄
UCS	C₁ ◁ C₂

1060 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 1060 conjugacy classes of subgroups — too many to draw. Only its 64 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) = 18
transitive degree	μ_t(G) = 120

A minimal faithful permutation representation — 3 generators on 18 points:

(1,3,2)(6,8)(10,11)(12,13)(14,15)(16,17);(3,4)(5,6,7,8)(9,10,12,13,11);(3,4)(15,16)(17,18)

Polycyclic presentation:

2^4,3,5^2,2^2|b^2=d,c^b=cd,e^a=e^2,e^b=e^2,f^a=f^4,f^c=f^4,g^a=f^3g,g^c=g^4,h^e=i,i^a=hi,i^b=hi,i^e=hi

Conjugacy classes & character table

133 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72260259
maximal normal subgroup of	#72260259

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, 2, 2, 3, 5, 5, 2, 2]);;
gap> SetPower(coll, 2, F.4);;
gap> SetConjugate(coll, 3, 2, F.3*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 5, 2, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.6^4);;
gap> SetConjugate(coll, 6, 3, F.6^4);;
gap> SetConjugate(coll, 7, 1, F.6^3*F.7);;
gap> SetConjugate(coll, 7, 3, F.7^4);;
gap> SetConjugate(coll, 8, 5, F.9);;
gap> SetConjugate(coll, 9, 1, F.8*F.9);;
gap> SetConjugate(coll, 9, 2, F.8*F.9);;
gap> SetConjugate(coll, 9, 5, F.8*F.9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);