#65721

order	320
isoclinism	isoclinic class of #3993
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₅×C₈ ⋊ C₄
Sylow	C₅ ⋊ #427

Subgroups

	subgroup	G/H
center	C₄	C₄⋊AGL(1,𝔽₅)
commutator	C₄×C₅	C₂²×C₄
Frattini	C₄	C₂²×AGL(1,𝔽₅)
Fitting	C₂×C₅×C₈	C₄
socle	C₂×C₅	C₂×C₄⋊C₄
prime radical	C₂²×D₅	D₄
cosocle	C₄×D₅	C₂³
hypercenter	C₂×C₈	AGL(1,𝔽₅)
hypocenter	C₅	#427
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈

114 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)(6,9,10,17,11,18,8,15)(7,13,12,16,14,20,19,21);(6,8,11,10)(7,12,14,19)(9,15,18,17)(13,16,20,21);(2,3,4,5)(6,7)(8,12)(9,16)(10,19)(11,14)(13,17)(15,20)(18,21)

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10598604, #10598605, #10598626, …
maximal normal subgroup of	#10598604, #10598605, #10598626, …
maximal quotient of	#10603814, #10603815, #10603820, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 1059);