#72227905

order	2400
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₂⁴×C₅

Aliases

No aliases for this group.

Splittings

Fitting, socle, derived	C₂⁴×C₅² ⋊ C₂×C₃
Hall	C₂⁴×C₅⋊D₅ ⋊ C₃
Sylow	C₅² ⋊ C₂×C₂²⋊A₄

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₂⁴×C₅²	C₂×C₃
LCS	G ▷ C₂⁴×C₅²
UCS	C₁

1044 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 60

A minimal faithful permutation representation — 3 generators on 23 points:

(1,2,4)(3,13,14,6,7,9)(5,10,15,11,12,8)(17,19,18)(21,23,22);(1,3,8,15,6)(4,9,12,10,14)(20,23)(21,22);(16,19)(17,18)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72227908, #72227911, #72227915
maximal normal subgroup of	#72227911

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