C₇×#13199 #13190415

order	1792
nilpotency class	3
isoclinism	isoclinic class of #2031
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#13199

Subgroups

	subgroup	G/H
center	C₇×C₈	#124
commutator	C₂²×C₄	C₂²×C₄×C₇
Frattini, cosocle	C₂²×C₈	C₂³×C₇
socle	C₂×C₇	#1955
prime radical	C₇×#575	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₇×C₈ ◁ C₂²×C₇×C₈ ◁ G

406 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 138 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)(8,9,13,19,14,20,35,49,15,21,36,50,12,18,23,38)(10,22,27,51,25,40,52,48,26,46,34,62,32,16,37,41)(11,17,30,44,29,43,58,67,31,45,61,70,33,47,53,64)(24,54,57,69,28,39,59,66,56,42,63,68,55,65,60,71);(8,10,23,52,12,32,36,27,15,26,35,37,14,25,13,34)(9,16,38,51,18,46,50,41,21,40,49,62,20,22,19,48)(11,28,58,57,33,24,30,60,31,55,53,63,29,56,61,59)(17,42,67,66,47,39,44,69,45,54,64,71,43,65,70,68);(8,11)(9,17)(10,24,26,56)(12,33)(13,30)(14,29)(15,31)(16,39,40,65)(18,47)(19,44)(20,43)(21,45)(22,42,46,54)(23,53)(25,28,32,55)(27,57,34,63)(35,58)(36,61)(37,60,52,59)(38,64)(41,66,48,71)(49,67)(50,70)(51,68,62,69)

Conjugacy classes & character table

364 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 6187);