C₅×#3185 #10616780

order	640
nilpotency class	3
isoclinism	isoclinic class of #3055
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₅×#3185

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₅
prime radical	C₂³×C₅	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ G

358 conjugacy classes of subgroups; only the 184 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 184 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)(8,11)(9,10)(12,13)(14,15,18,21)(16,22,23,29)(17,26,24,20)(19,27,28,25);(7,10)(11,13)(16,23)(19,28)(22,29)(25,27);(14,16,24,27,18,23,17,25)(15,19,26,29,21,28,20,22);(6,8,9,12)(7,11,10,13)(16,23)(19,28)(22,29)(25,27)

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#11996784, #12010152, #12075667, …
maximal subgroup of	C₅×#9416, C₅×#11838, C₅×#18557, …
maximal normal subgroup of	C₅×#9416, C₅×#11838, C₅×#18557, …
maximal quotient of	C₅×#20824, C₅×#21287, C₅×#21409, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(640, 18862);