#65370

order	320
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Splittings

Sylow

C₅ ⋊ #487

Subgroups

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₅
commutator, cosocle	C₂×C₄×C₅	C₂³
Frattini	C₂×C₄	C₂²×D₅
Fitting	C₅×C₄⋊Q₈	C₂
socle, prime radical	C₂²×C₅	C₂×D₄
hypercenter	C₄⋊Q₈	D₅
hypocenter	C₅	#487
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

98 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 3 generators on 21 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10599237, #10599240, #10599245, …
maximal normal subgroup of	#10599237, #10599240, #10599245, …
maximal quotient of	#10600365, #10600367, #10600375, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 708);