C₅×#12708 #11935428

order	1280
nilpotency class	4
isoclinism	isoclinic class of #12707
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₅×#12708

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#1781
commutator	C₂²×C₄	C₂²×C₄×C₅
Frattini, cosocle	C₂²×D₄	C₂³×C₅
prime radical	C₅×#3487	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂³×C₅ ◁ C₂²×C₅×#41 ◁ G

1002 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 1002 conjugacy classes of subgroups — too many to draw. Only its 146 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)(8,16,19,30)(9,13)(10,22,25,14)(11,15)(12,18)(17,33,35,32)(20,31,24,28)(21,29)(23,27)(26,34)(36,37);(6,8)(7,12)(9,17)(10,23)(11,19)(13,26)(14,30)(15,27)(16,22)(18,25)(20,36)(21,35)(24,34)(28,32)(29,37)(31,33);(6,9,11,21)(7,13,15,29)(8,17,19,35)(10,24,25,20)(12,26,27,37)(14,31,22,28)(16,32,30,33)(18,36,23,34)

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, 5696);