C₇×#9336 #13186552

order	1792
nilpotency class	3
isoclinism	isoclinic class of #9331
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#9336

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#386
commutator	C₂²×C₄	C₂²×C₄×C₇
Frattini, cosocle	C₂×C₄²	C₂³×C₇
prime radical	C₂³×C₇	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ G

360 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 114 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) = 896

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

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

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, 2324);