#72277107

order	2688
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

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

Subgroups

	subgroup	G/H
center, hypercenter	C₂	#1020⋊D₇
commutator, cosocle, hypocenter	C₇×#997	C₂²
Frattini	C₂³	D₇×S₄
Fitting	C₇×H(5,𝔽₂)	C₂×D₃
socle	C₂×C₇	#1020⋊C₂
LCS	G ▷ C₇×#997
UCS	C₁ ◁ C₂

386 conjugacy classes of subgroups; only the 17 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 17 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)(3,4)(5,6)(8,13,10,9)(11,15,14,12);(2,3)(4,5)(6,7)(9,14,13,12,10,15)

Polycyclic presentation:

2^2,3,7,2^5|c^b=c^2,d^a=d^6,d^b=d^6,e^a=ehi,e^b=ef,e^c=f,f^a=fgi,f^c=ef,g^c=gh,g^e=gi,h^b=gh,h^c=g,h^f=hi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72277119, D₇×#73026
maximal normal subgroup of	#72277119, D₇×#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");;
gap> coll := SingleCollector(F, [2, 2, 3, 7, 2, 2, 2, 2, 2]);;
gap> SetConjugate(coll, 3, 2, F.3^2);;
gap> SetConjugate(coll, 4, 1, F.4^6);;
gap> SetConjugate(coll, 4, 2, F.4^6);;
gap> SetConjugate(coll, 5, 1, F.5*F.8*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.6);;
gap> SetConjugate(coll, 5, 3, F.6);;
gap> SetConjugate(coll, 6, 1, F.6*F.7*F.9);;
gap> SetConjugate(coll, 6, 3, F.5*F.6);;
gap> SetConjugate(coll, 7, 3, F.7*F.8);;
gap> SetConjugate(coll, 7, 5, F.7*F.9);;
gap> SetConjugate(coll, 8, 2, F.7*F.8);;
gap> SetConjugate(coll, 8, 3, F.7);;
gap> SetConjugate(coll, 8, 6, F.8*F.9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);