#66262

order	320
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₅×Q₈ ⋊ C₄
Sylow	C₅ ⋊ #519

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂³×AGL(1,𝔽₅)
commutator, socle	C₂×C₅	C₂³×C₄
Fitting	C₂×C₅×Q₈	C₄
prime radical	C₂²×D₅	C₂³
cosocle	C₂×D₅	C₂⁴
hypercenter	C₂×Q₈	AGL(1,𝔽₅)
hypocenter	C₅	#519
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂ ◁ C₂×Q₈

266 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 80

A minimal faithful permutation representation — 4 generators on 21 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10603931, #10604563, #10604565, …
maximal normal subgroup of	#10603931, #10604563, #10604565, …
maximal quotient of	#10617093, #10617094, #10617095, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 1600);