C₇×#13669 #13190885

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

Aliases

direct product

C₇×#13669

Subgroups

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

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

normal subgroups only

This group has 288 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)(10,23)(11,28)(12,21)(13,37)(14,19)(15,22)(16,24)(17,25)(18,32)(20,30)(26,57)(27,50)(29,52)(31,40)(33,59)(34,60)(35,45)(36,53)(38,46)(39,51)(41,62)(42,61)(43,56)(44,58)(47,64)(48,65)(49,66)(54,55)(63,71)(67,69)(68,70);(8,10)(9,16)(11,29)(12,25)(13,38)(14,35)(15,27)(17,40)(18,46)(19,41)(20,52)(21,54)(22,43)(23,28)(24,32)(26,30)(31,61)(33,67)(34,63)(36,58)(37,42)(39,68)(44,48)(45,59)(47,57)(49,70)(50,60)(51,69)(53,71)(55,65)(56,66)(62,64);(8,10)(9,17)(11,24)(12,35)(13,26)(14,25)(15,27)(16,40)(18,42)(19,50)(20,44)(21,23)(22,45)(28,54)(29,32)(30,38)(31,62)(33,55)(34,56)(36,68)(37,46)(39,58)(41,60)(43,59)(47,71)(48,52)(49,69)(51,70)(53,57)(61,64)(63,66)(65,67)

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, 6657);