C₇×#11871 #13189087

order	1792
nilpotency class	4
isoclinism	isoclinic class of #11867
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#11871

Subgroups

	subgroup	G/H
center, socle	C₂×C₇	#1519
commutator	C₂×C₈	C₂²×C₄×C₇
Frattini, cosocle	C₂×M₁₆	C₂³×C₇
prime radical	C₂×C₇×#41	D₄
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₂×C₄×C₇ ◁ C₂×C₇×#131 ◁ G

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

normal subgroups only

This group has 360 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) = 71
transitive degree	μ_t(G) = 448

A minimal faithful permutation representation — 3 generators on 71 points:

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

Conjugacy classes & character table

238 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 4859);