#72209598

order	3000
isoclinism	isoclinic class of #72209598
automorphism	?
homology	Abelianization (H₁) = C₁ · Schur multiplier (H₂) = C₅

Aliases

No aliases for this group.

Splittings

Fitting, socle

C₅² ⋊ SL(2,𝔽₅)

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, hypocenter	G	C₁
Fitting, socle	C₅²	SL(2,𝔽₅)
cosocle	C₅⋊D₅	A₅
LCS	G
UCS	C₁

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Permutation generators:

(2,4,3,5)(6,11,9,8)(7,12,10,13)(14,23,18,20)(15,22,19,17)(16,24,21,25);(2,5,6)(3,4,9)(7,13,16)(8,19,14)(10,12,21)(11,15,18)(17,20,25)(22,23,24);(1,2,7,10,3)(4,8,17,14,6)(5,9,18,22,11)(12,20,25,16,15)(13,19,21,24,23);(1,4,12,13,5)(2,8,20,19,9)(3,6,15,23,11)(7,17,25,21,18)(10,14,16,24,22)

Conjugacy classes & character table

14 conjugacy classes — view the full character table.

Relationships

derived subgroup of	C₂²×#72209598, C₂²×#72210652, C₂²×AGL(2,𝔽₅), …
maximal subgroup of	#72210652, C₂×#72209598, #72227912, …
maximal normal subgroup of	#72210652, C₂×#72209598
maximal quotient of	#72209687, C₂×#72209598, #72227912, …

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its permutation generators:

gap> G := Group((2,4,3,5)(6,11,9,8)(7,12,10,13)(14,23,18,20)(15,22,19,17)(16,24,21,25), (2,5,6)(3,4,9)(7,13,16)(8,19,14)(10,12,21)(11,15,18)(17,20,25)(22,23,24), (1,2,7,10,3)(4,8,17,14,6)(5,9,18,22,11)(12,20,25,16,15)(13,19,21,24,23), (1,4,12,13,5)(2,8,20,19,9)(3,6,15,23,11)(7,17,25,21,18)(10,14,16,24,22));