#72217843

order	6048
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Hall	C₃²×ΓL(1,𝔽₈) ⋊ C₄×D₄
Sylow	C₇ ⋊ C₃×C₄×GO⁺(4,𝔽₂)

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	AGL(1,𝔽₇)×GO⁺(4,𝔽₂)
commutator	C₇×C₃⋊D₃	C₂²×C₃×C₄
Fitting, socle	C₂×C₃²×C₇	C₂×C₃×D₄
prime radical	C₂×#157⋊AGL(1,𝔽₇)	C₂
cosocle	C₂×C₇×C₃⋊D₃	C₂³×C₃
hypocenter	C₃²×C₇	C₃×C₄×D₄
LCS	G ▷ C₇×C₃⋊D₃ ▷ C₃²×C₇
UCS	C₁ ◁ C₂

656 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 83 normal subgroups are shown.

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

Full subgroup data →

Presentation

Polycyclic presentation:

2^3,3,2^2,3^2,7|b^2=e,c^a=cf,g^a=g^2,g^c=g^2h^2,g^f=g^2,h^a=g^2h,h^f=h^2,i^a=i^6,i^b=i^6,i^c=i^6,i^d=i^4

Conjugacy classes & character table

126 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₄×AGL(1,𝔽₇)×GO⁺(4,𝔽₂)
maximal normal subgroup of	C₄×AGL(1,𝔽₇)×GO⁺(4,𝔽₂)

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