#72276643

order	2688
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Sylow

C₇ ⋊ #87524

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₇×#1020
commutator, cosocle	C₂×C₇×C₂²⋊A₄	C₂²
Fitting, socle	C₂⁵×C₇	C₂×D₃
hypocenter	C₇×C₂²⋊A₄	D₄
LCS	G ▷ C₂×C₇×C₂²⋊A₄ ▷ C₇×C₂²⋊A₄
UCS	C₁ ◁ C₂

1652 conjugacy classes of subgroups; only the 34 normal subgroups are shown.

normal subgroups only

This group has 1652 conjugacy classes of subgroups — too many to draw. Only its 34 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) = 19
transitive degree	μ_t(G) = 84

A minimal faithful permutation representation — 2 generators on 19 points:

(2,3,4)(5,8,7)(10,12)(14,15)(16,17)(18,19);(1,2,4,3)(6,8)(9,10)(11,12)(13,14,16,18,19,17,15)

Polycyclic presentation:

2^3,3,7,2^4|b^a=bc,d^a=d^2,e^b=e^6,f^a=g,f^d=g,g^a=f,g^d=fg,h^a=hi,h^d=i,i^d=hi

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72276650, D₇×#87524
maximal normal subgroup of	#72276650, D₇×#87524

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