C₇×#8783 #13185999

order	1792
nilpotency class	3
isoclinism	isoclinic class of #8603
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#8783

Subgroups

	subgroup	G/H
center	C₂×C₄×C₇	#117
commutator	C₂²×C₄	C₂²×C₄×C₇
Frattini, cosocle	C₂×C₄²	C₂³×C₇
socle	C₂²×C₇	#382
prime radical	C₂³×C₇	#117
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₇×#376 ◁ G

262 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 126 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) = 55
transitive degree	μ_t(G) = 448

A minimal faithful permutation representation — 3 generators on 55 points:

(1,2,3,4,5,6,7)(8,9,12,15,11,14,19,22)(10,16,18,23,17,13,20,21)(24,25)(26,34,37,30)(27,38,40,53)(28,33)(29,32)(31,39,44,54)(35,48)(36,47,51,43)(41,45)(42,49)(46,55)(50,52);(8,10)(9,13)(11,17)(12,18)(14,16)(15,21)(19,20)(22,23)(24,26,28,36,29,37,41,51)(25,30,33,43,32,34,45,47)(27,35,39,49,40,50,54,55)(31,42,38,52,44,46,53,48);(10,17)(13,16)(18,20)(21,23)(24,27)(25,31,32,44)(26,35)(28,39)(29,40)(30,42,34,46)(33,38,45,53)(36,49)(37,50)(41,54)(43,52,47,48)(51,55)

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, 1771);