Finite Group Databasebeta

← Prev (#66266) Next (#66268) →

C₂³×#170 #66267

order	320
isoclinism	isoclinic class of AGL(1,𝔽₅)
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

C₂³×#170

Splittings

C₅ ⋊ C₂³×C₈

Subgroups

	subgroup	G/H
center, hypercenter	C₂⁴	AGL(1,𝔽₅)
commutator, hypocenter	C₅	C₂³×C₈
Frattini	C₂	C₂³×AGL(1,𝔽₅)
Fitting, socle, prime radical	C₂⁴×C₅	C₄
cosocle	Dic₅	C₂⁴
LCS	G ▷ C₅
UCS	C₁ ◁ C₂⁴

338 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 4 generators on 19 points:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10604223, C₂²⋊C₄×#170, #10604226, …
maximal normal subgroup of	#10604223, C₂²⋊C₄×#170, #10604226, …
maximal quotient of	C₂³×#674, C₂²×#3998, C₂²×C₄×#170, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 1605);