C₇×#10045 #13187261

order	1792
nilpotency class	3
isoclinism	isoclinic class of #10038
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#10045

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₇
prime radical	C₂³×C₇	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ G

444 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 98 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) = 896

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

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

Conjugacy classes & character table

217 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 3033);