#64717

order	320
isoclinism	isoclinic class of #3958
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Fitting	C₅×#108 ⋊ C₂
Sylow	C₅ ⋊ #359

Subgroups

	subgroup	G/H
center	C₄	#686
commutator	C₅×C₈	C₂×C₄
Frattini	C₂×C₈	C₂×D₅
Fitting	C₅×#108	C₂
socle	C₂×C₅	#102
prime radical	#4050	C₂
cosocle	C₂×C₅×C₈	C₂²
hypercenter	#108	D₅
hypocenter	C₅	#359
LCS	G ▷ C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #108

56 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 37
transitive degree	μ_t(G) = 160

A minimal faithful permutation representation — 2 generators on 37 points:

(1,2,4,5,3)(6,8,9,18,12,20,11,27)(7,13,14,28,16,30,26,29)(10,23,21,35,25,37,24,19)(15,31,22,17,32,33,36,34);(2,3)(4,5)(6,7)(8,17)(9,14)(10,22)(11,26)(12,16)(13,19)(15,24)(18,33)(20,34)(21,32)(23,28)(25,36)(27,31)(29,37)(30,35)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₅×#359, #10600034, #10600051, …
maximal normal subgroup of	D₅×#359, #10600034, #10600051, …
maximal quotient of	#10597958, #10597959, #10597962, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 55);