C₇×#13431 #13190647

order	1792
nilpotency class	4
isoclinism	isoclinic class of #13429
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#13431

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

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

normal subgroups only

This group has 402 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)(8,9,14,19)(10,20,24,41)(11,25,29,43)(12,30,33,17)(13,18,35,44)(15,31,38,22)(16,34,40,28)(21,47,26,56)(23,46,52,70)(27,58,60,55)(32,54,61,42)(36,65,67,71)(37,63,68,50)(39,59,62,64)(45,51,69,49)(48,66,57,53);(8,10,14,24)(9,15,19,38)(11,26,29,21)(12,31,33,22)(13,23,35,52)(16,36,40,67)(17,41,30,20)(18,37,44,68)(25,53,43,66)(27,48,60,57)(28,49,34,51)(32,63,61,50)(39,69,62,45)(42,70,54,46)(47,58,56,55)(59,65,64,71);(8,11,12,27,14,29,33,60)(9,16,17,39,19,40,30,62)(10,21,22,48,24,26,31,57)(13,34,32,64,35,28,61,59)(15,36,20,45,38,67,41,69)(18,43,42,58,44,25,54,55)(23,51,50,65,52,49,63,71)(37,53,46,56,68,66,70,47)

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, 6419);