#65353

order	320
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Sylow

C₅ ⋊ #499

Subgroups

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₅
commutator, cosocle	C₂×C₄×C₅	C₂³
Frattini	C₂×C₄	C₂²×D₅
Fitting	C₅×#125	C₂
socle, prime radical	C₂²×C₅	C₂×D₄
hypercenter	#125	D₅
hypocenter	C₅	#499
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #125

86 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 29
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 3 generators on 29 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10598744, #10598745, #10612887, …
maximal normal subgroup of	#10598744, #10598745, #10612887, …
maximal quotient of	#10600334, #10600342, #10600347, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 691);