C₅×#10602610 #72226425

order	3200
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₅×#10602610

Splittings

Sylow

C₅² ⋊ #1884

Subgroups

	subgroup	G/H
center	C₄×C₅	#4083
commutator	C₂×C₄×C₅	C₂²×C₄×C₅
Frattini	C₂²×C₄	C₂²×C₅×D₅
Fitting, prime radical	C₅²×#532	C₂
socle	C₂×C₅²	#386
cosocle	C₂²×C₄×C₅	C₂³×C₅
hypercenter	C₅×#532	D₅
hypocenter	C₅	C₅×#1884
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₄×C₅ ◁ C₂²×C₄×C₅ ◁ C₅×#532

974 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 80

A minimal faithful permutation representation — 3 generators on 26 points:

(1,2,3,4,5)(6,7,9,10,8)(11,13)(12,17)(14,20)(15,21)(16,23)(18,25)(19,24)(22,26);(12,16,18,19)(17,24,25,23);(7,8)(9,10)(11,12,14,19,15,18,22,16)(13,17,20,24,21,25,26,23)

Conjugacy classes & character table

440 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	#72226431
maximal normal subgroup of	#72226431

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((1,2,3,4,5)(6,7,9,10,8)(11,13)(12,17)(14,20)(15,21)(16,23)(18,25)(19,24)(22,26), (12,16,18,19)(17,24,25,23), (7,8)(9,10)(11,12,14,19,15,18,22,16)(13,17,20,24,21,25,26,23));

C5×#10602610 #72226425