C₇×#12168 #13189384

order	1792
nilpotency class	4
isoclinism	isoclinic class of #12108
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#12168

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#447
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₇
prime radical	C₂³×C₇	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂²×C₄×C₇ ◁ C₇×#424 ◁ G

330 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 74 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) = 79
transitive degree	μ_t(G) = 896

A minimal faithful permutation representation — 3 generators on 79 points:

(1,2,3,4,5,6,7)(12,15)(16,17,23,29)(18,30,34,58)(19,35,38,55)(20,39,43,53)(21,28,22,27)(24,46,49,32)(25,44,51,37)(26,41,54,42)(31,59,63,76)(33,64,45,52)(36,67,70,50)(40,56,71,48)(47,68,75,79)(57,62,78,61)(60,65,73,74)(66,72,77,69);(8,9)(10,12)(11,13)(14,15)(16,18,23,34)(17,24,29,49)(19,31,38,63)(20,40,43,71)(21,33,22,45)(25,47,51,75)(26,52,54,64)(27,48,28,56)(30,53,58,39)(32,41,46,42)(35,65,55,74)(36,68,70,79)(37,61,44,62)(50,76,67,59)(57,77,78,66)(60,72,73,69);(8,10,11,14)(9,12,13,15)(16,19)(17,25)(18,31)(20,36)(21,44)(22,37)(23,38)(24,47)(26,50)(27,55)(28,35)(29,51)(30,57)(32,60)(33,62)(34,63)(39,66)(40,68)(41,72)(42,69)(43,70)(45,61)(46,73)(48,74)(49,75)(52,76)(53,77)(54,67)(56,65)(58,78)(59,64)(71,79)

Conjugacy classes & character table

217 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 5156);