C₇×#12065 #13189281

order	1792
nilpotency class	4
isoclinism	isoclinic class of #2180
automorphism	?
homology	Abelianization (H₁) = C₂²×C₇×C₈ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#12065

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#442
commutator	C₈	C₂²×C₇×C₈
Frattini, cosocle	C₄×C₈	C₂³×C₇
prime radical	C₂×C₇×D₈	C₂×C₄
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₄²×C₇ ◁ C₇×#322 ◁ G

230 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 106 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) = 31
transitive degree	μ_t(G) = 896

A minimal faithful permutation representation — 3 generators on 31 points:

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

Conjugacy classes & character table

322 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 5053);