C₇×#12308 #13189524

order	1792
nilpotency class	4
isoclinism	isoclinic class of #12307
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#12308

Subgroups

	subgroup	G/H
center, socle	C₂×C₇	#1616
commutator, Frattini, cosocle	C₂×M₁₆	C₂³×C₇
prime radical	C₂³×C₇	AGL(1,ℤ/8)
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₂×C₄×C₇ ◁ C₂×C₇×M₁₆ ◁ G

274 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 70 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) = 39
transitive degree	μ_t(G) = 224

A minimal faithful permutation representation — 3 generators on 39 points:

(1,2,3,4,5,6,7)(8,9,12,15,13,18,29,17)(10,19,21,34,22,37,28,36)(11,23,26,16,27,33,25,35)(14,24,31,20,32,38,30,39);(8,10,13,22)(9,14,18,32)(11,24,27,38)(12,28,29,21)(15,30,17,31)(16,34,35,36)(19,23,37,33)(20,26,39,25);(9,15,18,17)(14,30,32,31)(16,23,35,33)(19,34,37,36)

Conjugacy classes & character table

175 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 5296);