#72246811

order	2016
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Hall	C₃²×C₇ ⋊ #100
Sylow	C₃² ⋊ #6407
Sylow	C₇ ⋊ #63559

Subgroups

	subgroup	G/H
center	C₂	#11759226
commutator	C₂²×C₃²×C₇	C₂×C₄
Frattini, hypercenter	C₂×C₄	(C₃⋊D₃)⋊D₇
Fitting, cosocle	C₂×C₃²×C₄×C₇	C₂²
socle	C₂×C₃²×C₇	C₂²⋊C₄
prime radical	C₂×C₇×C₄⋊(C₃⋊D₃)	C₂
hypocenter	C₃²×C₇	#100
LCS	G ▷ C₂²×C₃²×C₇ ▷ C₂×C₃²×C₇ ▷ C₃²×C₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄

238 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 59 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:

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

Conjugacy classes & character table

165 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72246813, D₇×#63559, #72246823, …
maximal normal subgroup of	#72246813, D₇×#63559, #72246823, …

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