#72259975

order	2400
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₃×C₄×C₅² ⋊ C₂²
Hall	C₃×C₅² ⋊ C₂×C₂²⋊C₄
Sylow	C₅² ⋊ C₂×(C₂×D₃)⋊C₄
Sylow	C₃ ⋊ #11717039

Subgroups

	subgroup	G/H
center, Frattini	C₂²	C₂×D₁₅⋊D₅
commutator	C₂×C₃×C₅²	C₂²×C₄
Fitting	C₂×C₃×C₄×C₅²	C₂²
socle, cosocle	C₂²×C₃×C₅²	C₂³
prime radical	C₂²×D₁₅⋊D₅	C₂
hypercenter	C₂×C₄	D₁₅⋊D₅
hypocenter	C₃×C₅²	C₂×C₂²⋊C₄
LCS	G ▷ C₂×C₃×C₅² ▷ C₃×C₅²
UCS	C₁ ◁ C₂² ◁ C₂×C₄

636 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 636 conjugacy classes of subgroups — too many to draw. Only its 103 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 240

Polycyclic presentation:

2^5,3,5^2|b^2=d,b^a=be,f^a=f^2,g^a=g^4,g^c=g^4,h^a=g^3h,h^c=h^4

Conjugacy classes & character table

204 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72260023, D₃×#11717039
maximal normal subgroup of	#72260023, D₃×#11717039
maximal quotient of	#72259987

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