#72277702

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

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₇×#120 ⋊ C₃×D₃
Sylow	C₇ ⋊ C₃×#73081

Subgroups

	subgroup	G/H
center, hypercenter	C₂	#72214620
commutator, cosocle	C₂×C₇×C₂²⋊A₄	C₂²×C₃
Frattini	C₂³	C₂×A₄⋊AGL(1,𝔽₇)
Fitting	C₂×C₇×#120	C₃×D₃
socle	C₂³×C₇	C₂×C₃×S₄
hypocenter	C₇×C₂²⋊A₄	C₃×D₄
LCS	G ▷ C₂×C₇×C₂²⋊A₄ ▷ C₇×C₂²⋊A₄
UCS	C₁ ◁ C₂

1026 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 1026 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) = 19
transitive degree	μ_t(G) = 84

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

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

Polycyclic presentation:

2^2,3,2,3,7,2^4|b^a=bd,e^b=e^2,f^b=f^6,f^c=f^2,g^a=gi,g^b=gh,g^e=h,h^a=hj,h^e=gh,i^b=ij,i^e=j,j^e=ij

Conjugacy classes & character table

86 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, 2, 3, 7, 2, 2, 2, 2]);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 5, 2, F.5^2);;
gap> SetConjugate(coll, 6, 2, F.6^6);;
gap> SetConjugate(coll, 6, 3, F.6^2);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.8);;
gap> SetConjugate(coll, 7, 5, F.8);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 5, F.7*F.8);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> SetConjugate(coll, 9, 5, F.10);;
gap> SetConjugate(coll, 10, 5, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);