C₇×#13671 #13190887

order	1792
nilpotency class	5
isoclinism	isoclinic class of #13669
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#13671

Subgroups

	subgroup	G/H
center, socle	C₂×C₇	#2184
commutator, Frattini, cosocle	C₂×C₁₆	C₂³×C₇
prime radical	C₇×#3415	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₂×C₄×C₇ ◁ C₇×#131 ◁ C₇×#504 ◁ G

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

normal subgroups only

This group has 284 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)(8,9,15,22)(10,23,27,52)(11,28,34,61)(12,21,14,19)(13,37,39,53)(16,40,44,24)(17,35,46,25)(18,47,51,32)(20,36,55,30)(26,58,59,45)(29,54,64,71)(31,41,33,60)(38,48,68,70)(42,57,56,63)(43,67,69,62)(49,66,50,65);(8,10)(9,16)(11,29)(12,25)(13,38)(14,35)(15,27)(17,41)(18,48)(19,42)(20,54)(21,56)(22,44)(23,28)(24,32)(26,30)(31,62)(33,67)(34,64)(36,59)(37,43)(39,68)(40,47)(45,50)(46,60)(49,58)(51,70)(52,61)(53,69)(55,71)(57,66)(63,65);(8,10)(9,17)(11,24)(12,35)(13,26)(14,25)(15,27)(16,41)(18,43)(19,52)(20,45)(21,23)(22,46)(28,56)(29,32)(30,38)(31,63)(33,57)(34,40)(36,68)(37,48)(39,59)(42,61)(44,60)(47,64)(49,71)(50,54)(51,69)(53,70)(55,58)(62,65)(66,67)

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, 6659);