C₇×#7832 #13185048

order	1792
nilpotency class	3
isoclinism	isoclinic class of #7734
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₇×#7832

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³×C₇	#120
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₇
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₂×C₄²×C₇ ◁ G

322 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 94 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) = 1792

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

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

Conjugacy classes & character table

259 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 820);