#72243171

order	3840
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Fitting	C₅×#379 ⋊ Dic₃
Sylow	C₅ ⋊ #11711334

Subgroups

	subgroup	G/H
center, hypercenter	C₂	#14512453
commutator	C₂×C₅×C₂²⋊A₄	C₂×C₄
Frattini	C₂³	C₂×A₄⋊AGL(1,𝔽₅)
Fitting	C₅×#379	Dic₃
socle	C₂³×C₅	C₂×SL(2,ℤ/4)
prime radical, cosocle	C₂×D₅×C₂²⋊A₄	C₂²
hypocenter	C₅×C₂²⋊A₄	C₄⋊C₄
LCS	G ▷ C₂×C₅×C₂²⋊A₄ ▷ C₅×C₂²⋊A₄
UCS	C₁ ◁ C₂

1516 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 1516 conjugacy classes of subgroups — too many to draw. Only its 43 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) = 60

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

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

Polycyclic presentation:

2^4,3,5,2^4|a^2=d,b^2=c,b^a=bd,e^b=e^2,f^b=f^3,f^c=f^4,g^a=gi,g^e=h,h^a=hj,h^b=gh,h^e=gh,i^e=j,j^b=ij,j^e=ij

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72243514, #72243722, AGL(1,𝔽₅)×#72991
maximal normal subgroup of	#72243514, #72243722, AGL(1,𝔽₅)×#72991

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