#72245771

order	2016
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Hall	C₃²×C₇ ⋊ C₂×#41
Sylow	C₃² ⋊ D₇×#41
Sylow	C₇ ⋊ #64306

Subgroups

	subgroup	G/H
center, hypercenter	C₄	D₃²×D₇
commutator, socle, cosocle	C₂×C₃²×C₇	C₂⁴
Frattini	C₂	C₂×D₃²×D₇
Fitting	C₃²×C₄×C₇	C₂³
hypocenter	C₃²×C₇	C₂×#41
LCS	G ▷ C₂×C₃²×C₇ ▷ C₃²×C₇
UCS	C₁ ◁ C₄

710 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 710 conjugacy classes of subgroups — too many to draw. Only its 142 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=e,b^2=e,c^2=e,d^2=e,c^a=ce,d^a=de,f^a=f^2,f^d=f^2,g^a=f^2g,g^d=g^2,h^c=h^6

Conjugacy classes & character table

138 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₇×#64306, #72246761, #72247182
maximal normal subgroup of	D₇×#64306, #72246761, #72247182

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