#72241663

order	6174
automorphism	?
homology	Abelianization (H₁) = C₂×C₃² · Schur multiplier (H₂) = C₃

Aliases

No aliases for this group.

Splittings

Fitting, socle, derived, Sylow	C₇³ ⋊ C₂×C₃²
Hall	ΓL(1,𝔽₈)×#3861 ⋊ C₂
Hall	C₇⋊(C₇⋊D₇) ⋊ C₃²

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₇³	C₂×C₃²
LCS	G ▷ C₇³
UCS	C₁

368 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 41 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) = 147

Polycyclic presentation:

2,3^2,7^3|d^a=d^6,d^c=d^2,e^a=e^6,e^c=e^2,f^a=f^6,f^b=f^2

Conjugacy classes & character table

61 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72241670, AGL(1,𝔽₇)×#64424, #72241692
maximal normal subgroup of	#72241670, AGL(1,𝔽₇)×#64424, #72241692

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