C₇×#12165 #13189381

order	1792
nilpotency class	4
isoclinism	isoclinic class of #12108
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#12165

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#447
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₇
prime radical	C₂²×C₇×D₈	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂²×C₄×C₇ ◁ C₇×#424 ◁ G

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

normal subgroups only

This group has 350 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) = 47
transitive degree	μ_t(G) = 448

A minimal faithful permutation representation — 3 generators on 47 points:

(1,2,3,4,5,6,7)(12,15)(17,23)(18,27)(19,30)(20,21)(24,38)(25,42)(26,37)(28,44)(29,34)(31,35)(32,33)(36,39)(40,47)(41,45);(8,9)(10,12)(11,13)(14,15)(16,17,22,26)(18,24,29,41)(19,31,33,42)(20,25,21,35)(23,32,37,30)(27,39,34,40)(28,45,44,38)(36,46,47,43);(8,10,11,14)(9,12,13,15)(16,18)(17,24)(19,28)(20,34)(21,27)(22,29)(23,36)(25,40)(26,41)(30,43)(31,45)(32,46)(33,44)(35,39)(37,47)(38,42)

Conjugacy classes & character table

217 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 5153);