#72246151

order	3840
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting	C₂²×C₅×H(5,𝔽₂) ⋊ D₃
derived	C₅×C₂²⋊#997 ⋊ C₂
Sylow	C₅ ⋊ #11715850

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	C₅⋊#87588
commutator, cosocle, hypocenter	C₅×C₂²⋊#997	C₂
Fitting	C₂²×C₅×H(5,𝔽₂)	D₃
socle	C₂³×C₅	#1020
LCS	G ▷ C₅×C₂²⋊#997
UCS	C₁ ◁ C₂

2378 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 2378 conjugacy classes of subgroups — too many to draw. Only its 39 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 — 3 generators on 17 points:

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

Polycyclic presentation:

2,3,5,2^7|b^a=b^2,c^a=c^4,d^a=e,d^b=e,e^a=d,e^b=de,f^a=g,f^b=g,g^a=f,g^b=fg,h^a=i,h^b=i,h^e=hj,i^a=h,i^b=hi,i^d=ij

Conjugacy classes & character table

85 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₅×#11715850
maximal normal subgroup of	D₅×#11715850
maximal quotient of	#72246164, #72253873

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", "j");;
gap> coll := SingleCollector(F, [2, 3, 5, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetConjugate(coll, 2, 1, F.2^2);;
gap> SetConjugate(coll, 3, 1, F.3^4);;
gap> SetConjugate(coll, 4, 1, F.5);;
gap> SetConjugate(coll, 4, 2, F.5);;
gap> SetConjugate(coll, 5, 1, F.4);;
gap> SetConjugate(coll, 5, 2, F.4*F.5);;
gap> SetConjugate(coll, 6, 1, F.7);;
gap> SetConjugate(coll, 6, 2, F.7);;
gap> SetConjugate(coll, 7, 1, F.6);;
gap> SetConjugate(coll, 7, 2, F.6*F.7);;
gap> SetConjugate(coll, 8, 1, F.9);;
gap> SetConjugate(coll, 8, 2, F.9);;
gap> SetConjugate(coll, 8, 5, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.8);;
gap> SetConjugate(coll, 9, 2, F.8*F.9);;
gap> SetConjugate(coll, 9, 4, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);