C₇×#10217 #13187433

order	1792
nilpotency class	3
isoclinism	isoclinic class of #10212
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#10217

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₇
prime radical	C₂³×C₇	#127
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ G

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

normal subgroups only

This group has 430 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) = 448

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

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

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 3205);