C₇×#13435 #13190651

order	1792
nilpotency class	4
isoclinism	isoclinic class of #13432
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#13435

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

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

normal subgroups only

This group has 408 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)(10,20,24,42)(11,25,29,54)(12,30,33,17)(13,34,35,18)(15,31,38,22)(16,39,41,28)(21,46,26,55)(23,50,51,45)(27,58,60,53)(32,43,64,62)(36,66,68,71)(37,49,65,63)(40,59,61,69)(44,57,70,48)(47,67,56,52);(8,10,14,24)(9,15,19,38)(11,26,29,21)(12,31,33,22)(13,23,35,51)(16,36,41,68)(17,42,30,20)(18,37,34,65)(25,52,54,67)(27,47,60,56)(28,48,39,57)(32,63,64,49)(40,70,61,44)(43,50,62,45)(46,58,55,53)(59,66,69,71);(8,11,12,27,14,29,33,60)(9,16,17,40,19,41,30,61)(10,21,22,47,24,26,31,56)(13,28,32,59,35,39,64,69)(15,36,20,44,38,68,42,70)(18,25,43,53,34,54,62,58)(23,48,49,71,51,57,63,66)(37,67,45,46,65,52,50,55)

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, 6423);