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C₄×#689 #65318

order	320
isoclinism	isoclinic class of #686
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

Splittings

C₅ ⋊ C₄×Q₁₆

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²⋊D₅
commutator	C₄×C₅	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₅
Fitting	C₄×C₅×Q₈	C₂
socle, prime radical	C₂²×C₅	C₂×D₄
cosocle	C₂×C₄×C₅	C₂³
hypercenter	C₄×Q₈	D₅
hypocenter	C₅	C₄×Q₁₆
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×Q₈

110 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 25
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 3 generators on 25 points:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10599006, #10599008, #10599011, …
maximal normal subgroup of	#10599006, #10599008, #10599011, …
maximal quotient of	#10600254, #10600258, C₄×#3942, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 656);