C₅×#1798 #10603306

order	640
nilpotency class	2
isoclinism	isoclinic class of #392
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#1798

Subgroups

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

312 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 136 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 — 3 generators on 29 points:

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

Conjugacy classes & character table

220 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#11986425, #11986428, #11991138, …
maximal subgroup of	C₅×#7252, C₅×#7253, C₅×#8409, …
maximal normal subgroup of	C₅×#7252, C₅×#7253, C₅×#8409, …
maximal quotient of	C₅×#8561, C₅×#8562, C₅×#8567, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(640, 5388);