#65184

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅×#108 ⋊ C₂
Sylow	C₅ ⋊ #443

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	C₄×D₄
prime radical	#4050	C₂
cosocle	C₂×C₄×C₅	C₂³
hypercenter	#108	D₅
hypocenter	C₅	#443
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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10598623, #10598624, #10600031, …
maximal normal subgroup of	#10598623, #10598624, #10600031, …
maximal quotient of	#10599309, #10599314, #10599315, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 522);