#72230517

order	15876
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂×C₃

Aliases

No aliases for this group.

Splittings

Fitting, socle, derived	C₃³×C₇² ⋊ C₂²×C₃
Hall	C₃³×#3861 ⋊ C₂²
Sylow	C₇² ⋊ C₃×D₃×C₃⋊D₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃³×C₇²	C₂²×C₃
LCS	G ▷ C₃³×C₇²
UCS	C₁

1466 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 1466 conjugacy classes of subgroups — too many to draw. Only its 88 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) = 23
transitive degree	μ_t(G) = 126

A minimal faithful permutation representation — 3 generators on 23 points:

(2,5,6,3,4,7)(8,9,10,14,11,13)(16,17)(19,20)(21,22);(1,2,4,7,6,5,3)(9,10)(11,12)(13,14)(18,19,20)(22,23);(15,16,17)

Polycyclic presentation:

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

Conjugacy classes & character table

180 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#72213608, #72230569, #72230589
maximal normal subgroup of	D₃×#72213608, #72230569, #72230589

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