#72229626

order	2400
automorphism	?
homology	Abelianization (H₁) = C₄×C₈ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Splittings

derived, Hall	C₃×C₅² ⋊ C₄×C₈
Sylow	C₅² ⋊ C₈×Dic₃
Sylow	C₃ ⋊ #11716761

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₄	#10597183
commutator, hypocenter	C₃×C₅²	C₄×C₈
Fitting	C₃×C₄×C₅²	C₂×C₄
socle	C₂×C₃×C₅²	C₄²
prime radical	C₂×C₃×C₅⋊D₅	C₂×C₄
cosocle	C₃×C₄×C₅⋊D₅	C₂²
LCS	G ▷ C₃×C₅²
UCS	C₁ ◁ C₄

236 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 53 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) = 21
transitive degree	μ_t(G) = 120

A minimal faithful permutation representation — 2 generators on 21 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

96 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#11716761, #72230521, #72230533, …
maximal normal subgroup of	D₃×#11716761, #72230521, #72230533, …
maximal quotient of	#72259968

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