C₇×#13657 #13190873

order	1792
nilpotency class	5
isoclinism	isoclinic class of #13657
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#13657

Subgroups

	subgroup	G/H
center, socle	C₂×C₇	#2177
commutator	C₁₆	C₂²×C₄×C₇
Frattini, cosocle	C₂×C₁₆	C₂³×C₇
prime radical	C₇×#508	C₂²
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₂×C₄×C₇ ◁ C₄×C₇×C₈ ◁ C₇×#346 ◁ G

182 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 182 conjugacy classes of subgroups — too many to draw. Only its 82 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)(10,22,25,56)(11,19)(12,30)(13,36)(14,39)(15,21)(16,24,42,63)(17,28)(18,33)(20,38)(23,58,60,71)(26,65,40,57)(27,66)(29,51)(31,50,59,55)(32,48)(34,45)(35,53)(37,54,62,46)(41,69,43,70)(44,68,52,64)(47,67)(49,61);(8,10)(9,16)(11,23)(12,31)(13,37)(14,40)(15,25)(17,41)(18,46)(19,50)(20,52)(21,42)(22,53)(24,61)(26,38)(27,64)(28,62)(29,60)(30,56)(32,68)(33,63)(34,70)(35,59)(36,43)(39,44)(45,65)(47,57)(48,58)(49,54)(51,55)(66,71)(67,69);(8,11,14,28,15,29,38,13)(9,17,20,19,21,36,39,51)(10,23,40,62,25,60,26,37)(12,32,34,33,35,27,67,61)(16,41,52,50,42,43,44,55)(18,47,48,53,49,45,66,30)(22,54,65,71,56,46,57,58)(24,31,68,70,63,59,64,69)

Conjugacy classes & character table

238 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 6645);