C₇×#12885 #13190101

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

Aliases

direct product

C₇×#12885

Subgroups

	subgroup	G/H
center	C₇×C₈	#124
commutator	C₂²×C₄	C₂²×C₄×C₇
Frattini, cosocle	C₂²×C₈	C₂³×C₇
socle	C₂×C₇	#1832
prime radical	C₂³×C₇	#131
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₇×C₈ ◁ C₂²×C₇×C₈ ◁ G

234 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 122 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)(8,9,13,19,14,20,37,52,15,21,38,53,12,18,34,49)(10,22,26,54,24,43,58,50,27,41,35,66,25,16,39,42)(11,28,32,61,31,48,64,51,33,46,36,67,30,17,44,47)(23,55,56,69,29,62,63,71,57,40,60,68,59,45,65,70);(8,10,12,25,15,27,14,24)(9,16,18,41,21,43,20,22)(11,29,31,57,33,59,30,23)(13,35,34,58,38,26,37,39)(17,45,28,55,48,62,46,40)(19,50,49,54,53,42,52,66)(32,65,64,56,36,63,44,60)(47,71,61,68,51,70,67,69);(8,11,12,30,15,33,14,31)(9,17,18,46,21,48,20,28)(10,23,24,29,27,57,25,59)(13,36,34,64,38,32,37,44)(16,40,22,45,43,55,41,62)(19,51,49,61,53,47,52,67)(26,60,58,65,35,56,39,63)(42,69,54,71,50,68,66,70)

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, 5873);