#65771

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

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₅×D₄ ⋊ C₄
Sylow	C₅ ⋊ #410

Subgroups

	subgroup	G/H
center	C₂	C₂×C₂²⋊AGL(1,𝔽₅)
commutator	C₂²×C₅	C₂²×C₄
Frattini	C₂²	C₂²×AGL(1,𝔽₅)
Fitting	C₂×C₅×D₄	C₄
socle	C₂×C₅	C₂×C₂²⋊C₄
prime radical	C₂×#711	C₂
cosocle	C₂²×D₅	C₂³
hypercenter	C₂×D₄	AGL(1,𝔽₅)
hypocenter	C₅	#410
LCS	G ▷ C₂²×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×D₄

158 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:

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

Polycyclic presentation:

2^6,5|a^2=d,b^2=f,c^a=ce,e^a=ef,g^a=g^2,c^b=cf,d^c=df,g^d=g^4

Conjugacy classes & character table

32 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10598569, #10603926, #10604458, …
maximal normal subgroup of	#10598569, #10603926, #10604458, …
maximal quotient of	#10604200, #10604203, #10604210, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 1109);