C₅×#2984 #10616579

order	640
nilpotency class	2
isoclinism	isoclinic class of D₄²
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄×C₅ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₅×#2984

Subgroups

	subgroup	G/H
center	C₂×C₄×C₅	C₂⁴
commutator	C₂²	C₂³×C₄×C₅
Frattini, cosocle	C₂×C₄	C₂⁴×C₅
socle	C₂²×C₅	C₂³×C₄
prime radical	C₂³×C₅	C₂²×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ G

364 conjugacy classes of subgroups; only the 272 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 272 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 29 points:

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

Conjugacy classes & character table

250 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#11995130, #11995144, #11995642, …
maximal subgroup of	C₅×#9164, C₅×#9166, C₅×#9244, …
maximal normal subgroup of	C₅×#9164, C₅×#9166, C₅×#9244, …
maximal quotient of	C₅×#20535, C₅×#20572, C₅×#20573, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(640, 18661);