C₇×#13666 #13190882

order	1792
nilpotency class	5
isoclinism	isoclinic class of #13661
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#13666

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	C₂×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂³×C₇
prime radical	C₂×C₇×#508	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂²×C₄×C₇ ◁ C₂²×C₇×C₈ ◁ C₂²×C₇×C₁₆ ◁ G

288 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 78 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) = 75
transitive degree	μ_t(G) = 896

A minimal faithful permutation representation — 3 generators on 75 points:

(1,2,3,4,5,6,7)(8,9)(10,11)(12,13,18,24)(14,25,30,56)(15,20,33,23)(16,34,37,68)(17,38,40,46)(19,41,44,27)(21,47,50,36)(22,51,53,32)(26,55,29,62)(28,60,61,74)(31,63,66,67)(35,52,70,75)(39,48,58,73)(42,59,54,57)(43,64,71,69)(45,72,49,65);(12,14)(13,19)(15,26)(16,35)(17,39)(18,30)(20,42)(21,48)(22,52)(23,54)(24,44)(25,34)(27,36)(28,32)(29,33)(31,64)(37,70)(38,43)(40,58)(41,47)(45,60)(46,71)(49,74)(50,73)(51,61)(53,75)(55,63)(56,68)(57,65)(59,72)(62,67)(66,69);(9,11)(12,15,18,33)(13,20,24,23)(14,26,30,29)(16,31,37,66)(17,32,40,51)(19,42,44,54)(21,45,50,49)(22,46,53,38)(25,55,56,62)(27,57,41,59)(28,58,61,39)(34,63,68,67)(35,64,70,69)(36,65,47,72)(43,52,71,75)(48,60,73,74)

Conjugacy classes & character table

343 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 6654);