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D₅×#108 #65181

order	320
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

Splittings

Fitting	C₅×#108 ⋊ C₂
Sylow	C₅ ⋊ C₂×#108

Subgroups

	subgroup	G/H
center	C₄	D₄×D₅
commutator	C₄×C₅	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₅
Fitting	C₅×#108	C₂
socle	C₂×C₅	C₂×C₄⋊C₄
prime radical	C₂²×D₅	Q₈
cosocle	C₂×C₄×C₅	C₂³
hypercenter	#108	D₅
hypocenter	C₅	C₂×#108
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ #108

106 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 80

A minimal faithful permutation representation — 3 generators on 21 points:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10598625, #10598626, #10598627, …
maximal normal subgroup of	#10598625, #10598626, #10598627, …
maximal quotient of	D₅×#334, #10599320, #10599322, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 519);