#65390

order	320
isoclinism	isoclinic class of Dic₁₀
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Splittings

Sylow

C₅ ⋊ #405

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂×D₅
commutator	C₂×C₅	C₂×C₄²
Frattini	C₂×C₄	C₂²×D₅
Fitting	C₂²×C₅×C₈	C₂
socle	C₂²×C₅	C₂²×C₄
prime radical	C₂³×C₅	C₂×C₄
cosocle	C₂×C₄×C₅	C₂³
hypercenter	C₂²×C₈	D₅
hypocenter	C₅	#405
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈

130 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 25
transitive degree	μ_t(G) = 160

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10598296, #10598304, #10598305, …
maximal normal subgroup of	#10598296, #10598304, #10598305, …
maximal quotient of	C₈×#681, #10600517, #10600518, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 728);