#72277810

order	2688
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

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

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₇×#120⋊C₃
commutator, cosocle	C₂⁵×C₇	C₂²×C₃
Frattini	C₂³	C₂×A₄×D₇
Fitting	C₇×#379	C₂×C₃
socle	C₂³×C₇	C₂²×A₄
hypocenter	C₂⁴×C₇	C₃×D₄
LCS	G ▷ C₂⁵×C₇ ▷ C₂⁴×C₇
UCS	C₁ ◁ C₂

1254 conjugacy classes of subgroups; only the 32 normal subgroups are shown.

normal subgroups only

This group has 1254 conjugacy classes of subgroups — too many to draw. Only its 32 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)(7,8,11)(9,10,12)(13,14)(15,16)(17,18);(1,2)(3,4)(5,12,11,6)(7,10,8,9)(14,15)(16,17)(18,19)

Polycyclic presentation:

2^2,3,2^5,7|b^a=bd,e^a=eg,e^c=ef,f^a=fh,f^c=e,g^c=gh,h^c=g,i^a=i^6,i^b=i^6

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72277843
maximal normal subgroup of	#72277843

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