C₇×#12437 #13189653

order	1792
nilpotency class	4
isoclinism	isoclinic class of #12436
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#12437

Subgroups

	subgroup	G/H
center, socle	C₂×C₇	#1667
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₇
prime radical	C₇×#135	D₄
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₂×C₄×C₇ ◁ C₄×C₇×C₈ ◁ G

214 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 74 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)(8,9,16,23)(10,24,30,54)(11,31,34,62)(12,20,36,48)(13,19,35,49)(14,22,15,21)(17,39,44,26)(18,45,47,32)(25,55,58,66)(27,43,59,42)(28,53,29,41)(33,63,38,50)(37,51,64,46)(40,57,67,61)(52,65,69,56)(60,70,71,68);(8,10,13,27,14,28,36,39,16,30,35,59,15,29,12,26)(9,17,20,41,21,42,49,54,23,44,48,53,22,43,19,24)(11,25,37,60,33,57,45,65,34,58,64,71,38,61,32,56)(18,40,50,68,46,66,62,69,47,67,63,70,51,55,31,52);(8,11)(9,18)(10,25)(12,32)(13,37)(14,33)(15,38)(16,34)(17,40)(19,31)(20,50)(21,46)(22,51)(23,47)(24,52)(26,56)(27,60)(28,57)(29,61)(30,58)(35,64)(36,45)(39,65)(41,68)(42,66)(43,55)(44,67)(48,63)(49,62)(53,70)(54,69)(59,71)

Conjugacy classes & character table

259 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 5425);