C₅×#10602934 #72226534

order	3200
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₅×#10602934

Splittings

Sylow

C₅² ⋊ #1795

Subgroups

	subgroup	G/H
center	C₂×C₅	#65536
commutator	C₂×C₄×C₅	C₂²×C₄×C₅
Frattini	C₂²×C₄	C₂²×C₅×D₅
Fitting, prime radical	C₂×C₅²×H(5,𝔽₂)	C₂
socle	C₂×C₅²	#379
cosocle	C₂²×C₄×C₅	C₂³×C₅
hypercenter	C₂×C₅×H(5,𝔽₂)	D₅
hypocenter	C₅	C₅×#1795
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₅ ◁ C₂²×C₄×C₅ ◁ C₂×C₅×H(5,𝔽₂)

2074 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)(15,22)(16,24)(18,25)(19,26)(21,23);(12,16)(14,23)(17,24)(18,19)(20,21)(25,26);(7,8)(9,10)(11,12,14,18,15,19,23,16)(13,17,21,24,22,26,20,25)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	#72226543
maximal normal subgroup of	#72226543

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

C5×#10602934 #72226534