C₅×#12710 #11935430

order	1280
nilpotency class	4
isoclinism	isoclinic class of #12707
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₅×#12710

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#1781
commutator	C₂²×C₄	C₂²×C₄×C₅
Frattini, cosocle	C₂²×Q₈	C₂³×C₅
prime radical	C₅×#3486	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂³×C₅ ◁ C₂³×C₅×Q₈ ◁ G

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

normal subgroups only

This group has 952 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)(6,7,11,16,12,17,9,14)(8,18,19,36,21,33,27,35)(10,23,22,34,26,15,25,32)(13,28,31,24,30,37,29,20);(6,8)(7,13)(9,19)(10,24)(11,27)(12,21)(14,29)(15,33)(16,31)(17,30)(18,23)(20,26)(22,28)(25,37)(32,35)(34,36);(6,9,12,11)(7,14,17,16)(8,19,21,27)(10,25,26,22)(13,29,30,31)(15,34,23,32)(18,35,33,36)(20,28,24,37)

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, 5698);