#64997

order	320
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₅×#97 ⋊ C₂
Sylow	C₅ ⋊ #405

Subgroups

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₅
commutator	C₂×C₅	C₂×C₄²
Fitting	C₅×#97	C₂
socle	C₂²×C₅	C₂²×C₄
prime radical	C₂²×D₅	C₂×C₄
cosocle	C₂×C₄×C₅	C₂³
hypercenter	#97	D₅
hypocenter	C₅	#405
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ #97

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10599880, #10599882, #10603868, …
maximal normal subgroup of	#10599880, #10599882, #10603868, …
maximal quotient of	#10598860, #10598861, #10598862, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 335);