C₇×#10211 #13187427

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

Aliases

direct product

C₇×#10211

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

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

normal subgroups only

This group has 406 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) = 71
transitive degree	μ_t(G) = 448

A minimal faithful permutation representation — 3 generators on 71 points:

(1,2,3,4,5,6,7)(8,9,16,22)(10,23,29,37)(11,30,34,25)(12,35,40,58)(13,41,46,54)(14,24,48,60)(15,21,44,57)(17,49,27,31)(18,51,55,39)(19,45,59,71)(20,38,61,64)(26,62,53,70)(28,52,63,69)(32,68,47,42)(33,66,67,36)(43,65,56,50);(8,10,15,28)(9,11,21,33)(12,36,39,25)(13,42,45,26)(14,17,38,50)(16,29,44,63)(18,52,35,23)(19,56,54,31)(20,47,60,70)(22,34,57,67)(24,62,61,32)(27,64,65,48)(30,40,66,51)(37,55,69,58)(41,49,59,43)(46,68,71,53);(8,11,15,33)(9,17,21,50)(10,24,28,61)(12,37,39,69)(13,43,45,49)(14,47,38,70)(16,34,44,67)(18,53,35,68)(19,36,54,25)(20,29,60,63)(22,27,57,65)(23,51,52,40)(26,58,42,55)(30,59,66,41)(31,46,56,71)(32,64,62,48)

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, 3199);