C₇×#13655 #13190871

order	1792
nilpotency class	5
isoclinism	isoclinic class of D₃₂
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#13655

Subgroups

	subgroup	G/H
center	C₇×C₈	D₁₆
commutator	C₁₆	C₂²×C₄×C₇
Frattini, cosocle	C₂×C₁₆	C₂³×C₇
socle	C₂×C₇	C₄×D₁₆
prime radical	C₇×#2262	C₂
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₇×C₈ ◁ C₂×C₇×C₈ ◁ C₄×C₇×C₈ ◁ C₄×C₇×C₁₆ ◁ G

186 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 86 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)(8,9,13,19,15,21,36,20)(10,22,25,52,27,53,26,58)(11,17,29,44,30,45,14,37)(12,31,33,61,35,63,67,62)(16,39,41,57,43,56,50,24)(18,28,47,59,49,60,48,34)(23,51,54,70,55,71,38,46)(32,40,64,68,66,69,65,42);(8,10)(9,16)(11,23)(12,32)(13,25)(14,38)(15,27)(17,40)(18,46)(19,41)(20,50)(21,43)(22,31)(24,34)(26,36)(28,39)(29,54)(30,55)(33,64)(35,66)(37,42)(44,68)(45,69)(47,51)(48,71)(49,70)(52,61)(53,63)(56,60)(57,59)(58,62)(65,67);(8,11,13,29,15,30,36,14)(9,17,19,44,21,45,20,37)(10,23,25,54,27,55,26,38)(12,28,33,59,35,60,67,34)(16,40,41,68,43,69,50,42)(18,31,47,61,49,63,48,62)(22,51,52,70,53,71,58,46)(24,32,39,64,57,66,56,65)

Conjugacy classes & character table

532 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 6643);