#72260345

order	4800
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Splittings

derived	C₅²×A₄ ⋊ C₄²
Sylow	C₅² ⋊ C₄×SL(2,ℤ/4)

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	#72213888
commutator, hypocenter	C₅²×A₄	C₄²
Fitting, socle	C₂³×C₅²	C₂×Dic₃
prime radical, cosocle	C₂×A₄×C₅⋊D₅	C₂²
LCS	G ▷ C₅²×A₄
UCS	C₁ ◁ C₂

626 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 626 conjugacy classes of subgroups — too many to draw. Only its 55 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) = 60

A minimal faithful permutation representation — 2 generators on 18 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72260358
maximal normal subgroup of	#72260358

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