#72277843

order	8064
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃² · Schur multiplier (H₂) = C₂³×C₃

Aliases

No aliases for this group.

Splittings

Fitting	C₇×#379 ⋊ C₂×C₃²
Hall	#11731306 ⋊ C₃²
Sylow	C₇ ⋊ C₃×AΣL(1,𝔽₆₄)

Subgroups

	subgroup	G/H
center, hypercenter	C₂	AGL(1,𝔽₇)×#120⋊C₃
commutator, cosocle	C₂⁵×C₇	C₂²×C₃²
Frattini	C₂³	C₂×A₄×AGL(1,𝔽₇)
Fitting	C₇×#379	C₂×C₃²
socle	C₂³×C₇	C₂²×C₃×A₄
hypocenter	C₂⁴×C₇	C₃²×D₄
LCS	G ▷ C₂⁵×C₇ ▷ C₂⁴×C₇
UCS	C₁ ◁ C₂

2644 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 2644 conjugacy classes of subgroups — too many to draw. Only its 65 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) = 19
transitive degree	μ_t(G) = 84

A minimal faithful permutation representation — 2 generators on 19 points:

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

Polycyclic presentation:

2^2,3^2,2^5,7|b^a=be,f^a=fh,f^d=g,g^a=gi,g^d=fg,h^d=i,i^d=hi,j^a=j^6,j^b=j^6,j^c=j^2

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

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