#72277044

order	8064
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Fitting	C₇×H(5,𝔽₂) ⋊ C₂×C₃×D₃
derived	C₇×#997 ⋊ C₂²×C₃
Sylow	C₇ ⋊ C₃×#73026

Subgroups

	subgroup	G/H
center, hypercenter	C₂	#72214629
commutator, cosocle, hypocenter	C₇×#997	C₂²×C₃
Frattini	C₂³	S₄×AGL(1,𝔽₇)
Fitting	C₇×H(5,𝔽₂)	C₂×C₃×D₃
socle	C₂×C₇	C₃×#1020⋊C₂
LCS	G ▷ C₇×#997
UCS	C₁ ◁ C₂

802 conjugacy classes of subgroups; only the 28 normal subgroups are shown.

normal subgroups only

This group has 802 conjugacy classes of subgroups — too many to draw. Only its 28 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) = 56

A minimal faithful permutation representation — 2 generators on 15 points:

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

Polycyclic presentation:

2^2,3^2,7,2^5|d^b=d^2,e^a=e^6,e^c=e^4,f^a=fij,f^b=fg,f^d=g,g^a=ghj,g^d=fg,h^d=hi,h^f=hj,i^b=hi,i^d=h,i^g=ij

Conjugacy classes & character table

73 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₇)×#73026
maximal normal subgroup of	AGL(1,𝔽₇)×#73026

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", "j");;
gap> coll := SingleCollector(F, [2, 2, 3, 3, 7, 2, 2, 2, 2, 2]);;
gap> SetConjugate(coll, 4, 2, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.5^6);;
gap> SetConjugate(coll, 5, 3, F.5^4);;
gap> SetConjugate(coll, 6, 1, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.7);;
gap> SetConjugate(coll, 6, 4, F.7);;
gap> SetConjugate(coll, 7, 1, F.7*F.8*F.10);;
gap> SetConjugate(coll, 7, 4, F.6*F.7);;
gap> SetConjugate(coll, 8, 4, F.8*F.9);;
gap> SetConjugate(coll, 8, 6, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.8*F.9);;
gap> SetConjugate(coll, 9, 4, F.8);;
gap> SetConjugate(coll, 9, 7, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);