C₇×#13664 #13190880

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

Aliases

direct product

C₇×#13664

Subgroups

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

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

normal subgroups only

This group has 324 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) = 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)(12,13,17,22)(14,18,26,33)(15,27,29,43)(16,30,28,21)(19,34,36,23)(20,37,38,24)(25,39,40,32)(31,42,35,41);(13,18)(14,23)(15,24)(16,28)(19,20)(21,39)(22,33)(25,42)(26,34)(27,31)(29,37)(30,32)(35,43)(36,38)(40,41);(9,11)

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, 6652);