C₇×#13686 #13190902

order	1792
nilpotency class	5
isoclinism	isoclinic class of #13661
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#13686

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	C₂×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂³×C₇
prime radical	C₇×#2246	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₄²×C₇ ◁ C₄×C₇×C₈ ◁ C₄×C₇×C₁₆ ◁ G

220 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 78 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) = 896

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

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

Conjugacy classes & character table

343 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 6674);