#65319

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

Aliases

No aliases for this group.

Splittings

Sylow

C₅ ⋊ #441

Subgroups

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₅
commutator	C₄×C₅	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₅
Fitting	C₄×C₅×Q₈	C₂
socle, prime radical	C₂²×C₅	C₂×D₄
cosocle	C₂×C₄×C₅	C₂³
hypercenter	C₄×Q₈	D₅
hypocenter	C₅	#441
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×Q₈

108 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 25
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 3 generators on 25 points:

(1,2,4,5,3)(6,7,8,9)(12,19)(15,24)(18,25)(20,22);(10,12,14,19)(11,15,17,24)(13,22,23,20)(16,25,21,18);(2,3)(4,5)(10,11,14,17)(12,18,19,25)(13,21,23,16)(15,22,24,20)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10611379, #10611381, #10611388, …
maximal normal subgroup of	#10611379, #10611381, #10611388, …
maximal quotient of	#10600251, #10600257, #10600268, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 657);