C₅×#11770 #11934490

order	1280
nilpotency class	4
isoclinism	isoclinic class of #11769
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₅×#11770

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#1505
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₂×M₁₆	C₂³×C₅
prime radical	C₅×#3176	C₂
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₅×#405 ◁ G

316 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 94 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) = 37
transitive degree	μ_t(G) = 160

A minimal faithful permutation representation — 3 generators on 37 points:

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

Conjugacy classes & character table

170 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 4758);