C₇×#13668 #13190884

order	1792
nilpotency class	5
isoclinism	isoclinic class of #13667
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#13668

Subgroups

	subgroup	G/H
center, socle	C₂×C₇	#2184
commutator, Frattini, cosocle	C₂×C₁₆	C₂³×C₇
prime radical	C₇×#3414	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₂×C₄×C₇ ◁ C₂²×C₇×C₈ ◁ C₂×C₇×M₃₂ ◁ G

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

normal subgroups only

This group has 284 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) = 71
transitive degree	μ_t(G) = 448

A minimal faithful permutation representation — 3 generators on 71 points:

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

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, 6656);