C₅×#13054 #11935774

order	1280
nilpotency class	4
isoclinism	isoclinic class of #2204
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₅×#13054

Subgroups

	subgroup	G/H
center	C₄×C₅	B(4,𝔽₂)
commutator	C₂×Q₈	C₂²×C₄×C₅
Frattini, cosocle	C₂×#41	C₂³×C₅
socle	C₂×C₅	#1889
prime radical	C₅×#3018	C₂
LCS	G ▷ C₂×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅ ◁ C₂×C₄×C₅ ◁ C₂×C₅×#41 ◁ G

632 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 632 conjugacy classes of subgroups — too many to draw. Only its 114 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)(8,17)(9,14,18,27)(10,15)(12,22)(13,25)(19,32)(20,29)(21,34,24,37)(23,31)(26,33)(28,35,30,36);(6,8,18,20,11,19,9,22)(7,12,27,17,16,29,14,32)(10,24,26,36,25,21,23,35)(13,30,31,37,15,28,33,34);(7,13)(8,19)(10,25)(12,28)(14,33)(15,16)(17,34)(20,22)(23,26)(27,31)(29,30)(32,37)

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, 6042);