C₇×#13441 #13190657

order	1792
nilpotency class	4
isoclinism	isoclinic class of #13313
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#13441

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#466
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₇
prime radical	C₂³×C₇	#135
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂²×C₄×C₇ ◁ C₇×#405 ◁ G

378 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 110 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) = 75
transitive degree	μ_t(G) = 896

A minimal faithful permutation representation — 3 generators on 75 points:

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

Conjugacy classes & character table

280 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 6429);