C₇×#13438 #13190654

order	1792
nilpotency class	4
isoclinism	isoclinic class of #13436
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#13438

Subgroups

	subgroup	G/H
center, socle	C₂×C₇	#2044
commutator, Frattini, cosocle	C₂×#41	C₂³×C₇
prime radical	C₂×C₇×#41	C₂³
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₂×C₄×C₇ ◁ C₂×C₇×#41 ◁ G

406 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 86 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)(8,9,15,20)(10,21,26,46)(11,27,32,60)(12,33,14,18)(13,35,37,19)(16,25,41,23)(17,42,45,30)(22,51,54,71)(24,56,36,50)(28,61,52,67)(29,59,31,58)(34,47,38,49)(39,62,68,69)(40,55,48,57)(43,66,70,53)(44,63,65,64);(8,10,15,26)(9,16,20,41)(11,28,32,52)(12,25,14,23)(13,36,37,24)(17,43,45,70)(18,46,33,21)(19,48,35,40)(22,29,54,31)(27,51,60,71)(30,62,42,69)(34,55,38,57)(39,65,68,44)(47,50,49,56)(53,63,66,64)(58,67,59,61);(8,11,12,29,15,32,14,31)(9,17,18,44,20,45,33,65)(10,22,23,52,26,54,25,28)(13,30,34,63,37,42,38,64)(16,39,21,43,41,68,46,70)(19,27,47,58,35,60,49,59)(24,53,55,69,36,66,57,62)(40,67,50,51,48,61,56,71)

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 6426);