C₇×#10210 #13187426

order	1792
nilpotency class	3
isoclinism	isoclinic class of #10208
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#10210

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₇
prime radical	C₇×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ G

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

normal subgroups only

This group has 440 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) = 448

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

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

Conjugacy classes & character table

175 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 3198);