C₇×#13198 #13190414

order	1792
nilpotency class	3
isoclinism	isoclinic class of #2010
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#13198

Subgroups

	subgroup	G/H
center	C₇×C₈	#127
commutator	C₂²×C₄	C₂²×C₄×C₇
Frattini, cosocle	C₂²×C₈	C₂³×C₇
socle	C₂×C₇	#1955
prime radical	C₇×#575	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₇×C₈ ◁ C₂²×C₇×C₈ ◁ G

410 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 138 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)(8,9,12,17,13,18,28,39,14,19,29,40,30,41,54,65)(10,15,22,33,23,34,48,60,21,32,46,58,47,59,69,71)(11,16,26,37,27,38,43,56,25,36,50,62,53,64,68,66)(20,42,45,67,24,49,52,70,44,31,55,57,51,35,61,63);(8,10,12,22,13,23,28,48,14,21,29,46,30,47,54,69)(9,15,17,33,18,34,39,60,19,32,40,58,41,59,65,71)(11,24,50,61,27,44,68,45,25,51,26,52,53,20,43,55)(16,35,62,70,38,42,66,57,36,49,37,63,64,31,56,67);(8,11)(9,16)(10,20,21,44)(12,26)(13,27)(14,25)(15,31,32,42)(17,37)(18,38)(19,36)(22,45,46,55)(23,24,47,51)(28,43)(29,50)(30,53)(33,57,58,67)(34,35,59,49)(39,56)(40,62)(41,64)(48,52,69,61)(54,68)(60,63,71,70)(65,66)

Conjugacy classes & character table

364 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 6186);