C₅×C₇×#137 #11765307

order	1120
nilpotency class	3
isoclinism	isoclinic class of AGL(1,ℤ/8)
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅×C₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×C₇×#137

Subgroups

	subgroup	G/H
center, socle	C₂×C₅×C₇	C₂×D₄
commutator, Frattini, cosocle	C₄	C₂³×C₅×C₇
prime radical	C₅×C₇×#41	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅×C₇ ◁ C₂×C₄×C₅×C₇ ◁ G

120 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 120 conjugacy classes of subgroups — too many to draw. Only its 80 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) = 28
transitive degree	μ_t(G) = 560

A minimal faithful permutation representation — 3 generators on 28 points:

(1,2,3,4,5)(6,7,8,9,10,11,12)(13,14)(15,21)(16,24)(17,19)(18,22)(20,26)(23,28)(25,27);(13,15,17,23)(14,18,19,27)(16,25,26,22)(20,28,24,21);(14,19)(18,27)(20,24)(21,28)

Conjugacy classes & character table

385 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1120, 913);