C₇×#13313 #13190529

order	1792
nilpotency class	4
isoclinism	isoclinic class of #13313
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₇×#13313

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#466
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₇
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂²×C₄×C₇ ◁ C₂×C₇×#131 ◁ G

556 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 118 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) = 43
transitive degree	μ_t(G) = 448

A minimal faithful permutation representation — 3 generators on 43 points:

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

Conjugacy classes & character table

280 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 6301);