C₅×#12778 #11935498

order	1280
nilpotency class	4
isoclinism	isoclinic class of #12778
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

direct product

C₅×#12778

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#1790
commutator	C₂×Q₈	C₂²×C₄×C₅
Frattini, cosocle	C₂×#41	C₂³×C₅
prime radical	C₅×#3549	C₂
LCS	G ▷ C₂×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₂×C₅×#41 ◁ G

860 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 860 conjugacy classes of subgroups — too many to draw. Only its 138 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) = 21
transitive degree	μ_t(G) = 80

A minimal faithful permutation representation — 3 generators on 21 points:

(1,2,3,4,5)(6,7)(8,16,18,20)(9,13,19,12)(10,14)(11,15)(17,21);(7,12)(9,19)(10,17)(13,15)(14,16)(20,21);(6,8)(7,12)(9,17)(10,19)(11,18)(13,15)(14,20)(16,21)

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, 5766);