#72254358

order	4704
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Hall	C₃×C₇² ⋊ H(5,𝔽₂)
Sylow	C₇² ⋊ #1004
Sylow	C₃ ⋊ #13096712

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₃×D₇²
commutator, socle, cosocle	C₂×C₃×C₇²	C₂⁴
Fitting	C₂²×C₃×C₇²	C₂³
hypercenter	C₂²	D₃×D₇²
hypocenter	C₃×C₇²	H(5,𝔽₂)
LCS	G ▷ C₂×C₃×C₇² ▷ C₃×C₇²
UCS	C₁ ◁ C₂ ◁ C₂²

826 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 826 conjugacy classes of subgroups — too many to draw. Only its 140 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 168

Polycyclic presentation:

2^5,3,7^2|b^a=be,c^a=ce,d^b=de,f^d=f^2,g^a=g^6,g^b=g^6,h^a=g^2h,h^b=h^6

Conjugacy classes & character table

225 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72269515
maximal normal subgroup of	#72269515

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