#72244693

order	4032
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Fitting	C₂²×C₄×C₇ ⋊ C₂×C₃×D₃
Sylow	C₇ ⋊ C₃×D₄×S₄

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×S₄×AGL(1,𝔽₇)
commutator, cosocle	C₂×C₇×A₄	C₂³×C₃
Fitting	C₂²×C₄×C₇	C₂×C₃×D₃
socle	C₂³×C₇	C₂²×C₃×D₃
hypercenter	C₄	S₄×AGL(1,𝔽₇)
hypocenter	C₇×A₄	C₂×C₃×D₄
LCS	G ▷ C₂×C₇×A₄ ▷ C₇×A₄
UCS	C₁ ◁ C₂ ◁ C₄

968 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 968 conjugacy classes of subgroups — too many to draw. Only its 74 normal subgroups are shown.

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

Full subgroup data →

Presentation

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 84

A minimal faithful permutation representation — 3 generators on 15 points:

(1,2)(3,4)(5,7,8,6)(9,10,15)(11,12,14);(2,4)(10,11)(12,14)(13,15);(2,4)(7,8)

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₄×S₄×AGL(1,𝔽₇)
maximal normal subgroup of	D₄×S₄×AGL(1,𝔽₇)

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