C₅×#7390 #11930110

order	1280
nilpotency class	5
isoclinism	isoclinic class of #7389
automorphism	?
homology	Abelianization (H₁) = C₂×C₅×C₈ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅×#7390

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#1333
commutator	C₁₆	C₂×C₅×C₈
Frattini, cosocle	#346	C₂²×C₅
prime radical	C₂×C₅×D₁₆	C₄
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₄×C₅×C₈ ◁ C₅×#346 ◁ G

128 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 128 conjugacy classes of subgroups — too many to draw. Only its 50 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 — 2 generators on 37 points:

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

Conjugacy classes & character table

170 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	D₅×#7390, AGL(1,𝔽₅)×#7390, #72242243, …
maximal subgroup of	D₅×#7390, C₅×AGL(1,ℤ/32), #72242243
maximal normal subgroup of	D₅×#7390, C₅×AGL(1,ℤ/32), #72242243

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 378);