C₇×#13634 #13190850

order	1792
nilpotency class	5
isoclinism	isoclinic class of #2263
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#13634

Subgroups

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

194 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 194 conjugacy classes of subgroups — too many to draw. Only its 126 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)(8,9,10,11)(13,18)(14,19)(15,23)(16,26)(20,28)(21,37)(22,38)(24,40)(25,34)(27,39)(29,30)(31,36)(32,33)(35,43)(41,42);(12,13,19,30,15,21,32,38,26,36,40,43,39,41,25,28,17,20,34,42,27,35,24,31,16,22,33,37,23,29,14,18);(13,20)(18,28)(21,35)(22,36)(29,41)(30,42)(31,38)(37,43)

Conjugacy classes & character table

364 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 6622);