C₅×#2979 #10616574

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×#2979

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	C₅×#532	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ G

410 conjugacy classes of subgroups; only the 256 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 256 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)(8,12)(9,11)(10,13)(14,15,17,20,18,21,24,27)(16,19,23,26,22,25,28,29);(6,8)(7,10)(9,13)(11,12)(14,16)(15,19)(17,23)(18,22)(20,26)(21,25)(24,28)(27,29);(8,13)(10,12)(14,16)(15,19)(17,23)(18,22)(20,26)(21,25)(24,28)(27,29);(7,11)(10,12)(16,22)(19,25)(23,28)(26,29)

Conjugacy classes & character table

250 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#11993803, #11993808, #11994611, …
maximal subgroup of	C₅×#8934, C₅×#8935, C₅×#9067, …
maximal normal subgroup of	C₅×#8934, C₅×#8935, C₅×#9067, …
maximal quotient of	C₅×#20530, C₅×#20538, C₅×#20562, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(640, 18656);