#72224613

order	7200
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Splittings

Fitting, socle, derived, Hall	C₃²×C₅² ⋊ C₂×C₄²
Sylow	C₅² ⋊ C₄×#624
Sylow	C₃² ⋊ C₂×AGL(1,𝔽₅)²

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₃²×C₅²	C₂×C₄²
prime radical	D₅²×C₃⋊D₃	C₂²
cosocle	C₃²×D₅²	C₂³
LCS	G ▷ C₃²×C₅²
UCS	C₁

1020 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 1020 conjugacy classes of subgroups — too many to draw. Only its 129 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) = 16
transitive degree	μ_t(G) = 60

A minimal faithful permutation representation — 3 generators on 16 points:

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

Polycyclic presentation:

2^5,3^2,5^2|a^2=d,c^2=e,f^a=f^2,f^b=f^2,g^a=f^2g,g^b=g^2,h^a=h^3,h^c=h^3,h^d=h^4,h^e=h^4,i^a=h^3i,i^c=i^3,i^d=h^2i,i^e=i^4

Conjugacy classes & character table

114 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃²×AGL(1,𝔽₅)², C₂×#72224613
maximal normal subgroup of	D₃²×AGL(1,𝔽₅)², C₂×#72224613
maximal quotient of	#72231176, #72231207, #72231242, …
holomorph of	ΓL(1,𝔽₁₆)

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