#72231257

order	14400
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₂²×C₃

Aliases

No aliases for this group.

Splittings

Fitting, socle	C₂⁴×C₅² ⋊ C₃⋊Dic₃
derived	C₅²×A₄² ⋊ C₄
Sylow	C₅² ⋊ A₄⋊SL(2,ℤ/4)

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, hypocenter	C₅²×A₄²	C₄
Fitting, socle	C₂⁴×C₅²	C₃⋊Dic₃
prime radical, cosocle	A₄²×C₅⋊D₅	C₂
LCS	G ▷ C₅²×A₄²
UCS	C₁

1528 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 1528 conjugacy classes of subgroups — too many to draw. Only its 100 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) = 18
transitive degree	μ_t(G) = 300

A minimal faithful permutation representation — 3 generators on 18 points:

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

Polycyclic presentation:

2^2,3^2,5^2,2^4|a^2=b,c^a=c^2,d^a=d^2,e^a=e^3,e^b=e^4,f^a=f^3,f^b=f^4,g^c=h,g^d=gh,h^a=gh,h^c=gh,h^d=g,i^a=gi,i^c=gi,i^d=j,j^a=hij,j^c=ghj,j^d=ij

Conjugacy classes & character table

124 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#1043×A₄⋊S₄, #72260373
maximal normal subgroup of	#1043×A₄⋊S₄, #72260373

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