#72245756

order	2352
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Hall	(C₂×D₇)⋊Dic₇ ⋊ C₃
Hall	C₇×ΓL(1,𝔽₈) ⋊ C₂²⋊C₄
Sylow	C₇² ⋊ C₃×C₂²⋊C₄

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂²	D₇×AGL(1,𝔽₇)
commutator	C₂×C₇²	C₂×C₃×C₄
Fitting, socle, cosocle	C₂²×C₇²	C₂²×C₃
prime radical	C₂²×D₇×ΓL(1,𝔽₈)	C₂
hypocenter	C₇²	C₃×C₂²⋊C₄
LCS	G ▷ C₂×C₇² ▷ C₇²
UCS	C₁ ◁ C₂²

144 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 144 conjugacy classes of subgroups — too many to draw. Only its 47 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) = 336

Polycyclic presentation:

2^2,3,2^2,7^2|b^2=d,b^a=be,f^b=f^6,f^c=f^4,g^a=f^2g^6,g^b=g^6,g^c=f^3g

Conjugacy classes & character table

122 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72245767, #72245916, #72245937
maximal normal subgroup of	#72245767, #72245916, #72245937
maximal quotient of	#72245758, #72246115

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