#72243722

order	7680
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₅×#379 ⋊ C₄×D₃
Sylow	C₅ ⋊ #72243707

Subgroups

	subgroup	G/H
center, hypercenter	C₂	AGL(1,𝔽₅)×#1020⋊C₂
commutator	C₂×C₅×C₂²⋊A₄	C₂²×C₄
Frattini	C₂³	C₂×AGL(1,𝔽₅)×S₄
Fitting	C₅×#379	C₄×D₃
socle	C₂³×C₅	C₂×C₄×S₄
prime radical	C₂×D₅×#1020	C₂²
cosocle	C₂×D₅×C₂²⋊A₄	C₂³
hypocenter	C₅×C₂²⋊A₄	C₂×C₄⋊C₄
LCS	G ▷ C₂×C₅×C₂²⋊A₄ ▷ C₅×C₂²⋊A₄
UCS	C₁ ◁ C₂

5000 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 5000 conjugacy classes of subgroups — too many to draw. Only its 82 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 — 3 generators on 17 points:

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

Polycyclic presentation:

2^5,3,5,2^4|a^2=d,c^2=d,c^a=ce,f^b=f^2,g^a=g^3,g^c=g^3,g^d=g^4,h^a=hj,h^b=hi,h^f=i,i^a=ik,i^f=hi,j^b=jk,j^f=k,k^f=jk

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₅)×#11713200
maximal normal subgroup of	AGL(1,𝔽₅)×#11713200

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", "k");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 3, 5, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 3, F.4);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 6, 2, F.6^2);;
gap> SetConjugate(coll, 7, 1, F.7^3);;
gap> SetConjugate(coll, 7, 3, F.7^3);;
gap> SetConjugate(coll, 7, 4, F.7^4);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 8, 6, F.9);;
gap> SetConjugate(coll, 9, 1, F.9*F.11);;
gap> SetConjugate(coll, 9, 6, F.8*F.9);;
gap> SetConjugate(coll, 10, 2, F.10*F.11);;
gap> SetConjugate(coll, 10, 6, F.11);;
gap> SetConjugate(coll, 11, 6, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);