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D₅×#127 #66048

order	320
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

Splittings

Fitting	C₅×#127 ⋊ C₂
Sylow	C₅ ⋊ C₂×#127

Subgroups

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₅
commutator, socle, cosocle	C₂²×C₅	C₂⁴
Fitting	C₅×#127	C₂
hypercenter	#127	D₅
hypocenter	C₅	C₂×#127
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ #127

498 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 13
transitive degree	μ_t(G) = 80

A minimal faithful permutation representation — 4 generators on 13 points:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₅×#331, #10599176, #10599180, …
maximal normal subgroup of	D₅×#331, #10599176, #10599180, …
maximal quotient of	#10606050, #10606059, #10606065, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 1386);