#72260366

order	7200
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Fitting, socle	C₂²×C₃×C₅² ⋊ C₂×Dic₃
derived	C₃×C₅²×A₄ ⋊ C₂×C₄
Sylow	C₅² ⋊ D₃×SL(2,ℤ/4)

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, hypocenter	C₃×C₅²×A₄	C₂×C₄
Fitting, socle	C₂²×C₃×C₅²	C₂×Dic₃
prime radical	A₄×D₁₅⋊D₅	C₂
cosocle	C₃×A₄×C₅⋊D₅	C₂²
LCS	G ▷ C₃×C₅²×A₄
UCS	C₁

628 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 628 conjugacy classes of subgroups — too many to draw. Only its 49 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) = 17
transitive degree	μ_t(G) = 120

A minimal faithful permutation representation — 2 generators on 17 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

78 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72260373
maximal quotient of	#72260373

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, 3, 3, 5, 5, 2, 2]);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 4, 2, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.4^2*F.5);;
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.6^2*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);