C₅×#7216 #11929936

order	1280
nilpotency class	4
isoclinism	isoclinic class of #7213
automorphism	?
homology	Abelianization (H₁) = C₄²×C₅ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₅×#7216

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#342
commutator	C₂²×C₄	C₄²×C₅
Frattini, cosocle	C₂×#117	C₂²×C₅
prime radical	C₂⁴×C₅	C₄⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅ ◁ C₂×C₅×#117 ◁ G

276 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 74 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) = 45
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 2 generators on 45 points:

(1,2,3,4,5)(6,7)(8,10)(9,12)(11,13)(14,15,18,23,20,25,19,24)(16,26,28,41,30,43,29,42)(17,31,32,40,33,22,34,38)(21,35,36,27,37,44,39,45);(7,10,12,11)(14,16,19,29,20,30,18,28)(15,21,24,36,25,37,23,39)(17,27,32,44,33,45,34,35)(22,26,38,42,31,43,40,41)

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 204);