#72244351

order	2304
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Fitting

C₂³×Q₈ ⋊ D₃²

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×S₄²
commutator, cosocle, hypocenter	A₄×SL(2,𝔽₃)	C₂³
Fitting	C₂³×Q₈	D₃²
socle	C₂³	C₂×D₃×S₄
hypercenter	C₂²	S₄²
LCS	G ▷ A₄×SL(2,𝔽₃)
UCS	C₁ ◁ C₂ ◁ C₂²

1640 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 1640 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

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 48

A minimal faithful permutation representation — 3 generators on 20 points:

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

Polycyclic presentation:

2^3,3^2,2^5|a^2=j,h^2=j,i^2=j,b^a=bj,d^a=d^2,e^c=e^2,f^d=g,g^a=fg,g^d=fg,h^c=hj,h^e=i,i^c=hi,i^e=hij,i^h=ij

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal quotient of

GL(2,𝔽₃)×GL(2,ℤ/4)

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