C₇×#7831 #13185047

order	1792
nilpotency class	3
isoclinism	isoclinic class of #7734
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₇×#7831

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³×C₇	#120
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₇
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₂×C₄²×C₇ ◁ G

350 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 98 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) = 1792

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

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

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, 819);