C₇×#13654 #13190870

order	1792
nilpotency class	5
isoclinism	isoclinic class of #2263
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#13654

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	C₂×D₁₆
commutator	C₁₆	C₂²×C₄×C₇
Frattini, cosocle	C₂×C₁₆	C₂³×C₇
prime radical	C₂×C₇×D₃₂	C₂
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₄²×C₇ ◁ C₄×C₇×C₈ ◁ C₄×C₇×C₁₆ ◁ G

212 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 90 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)(8,9,11,10)(12,13,15,19)(14,18,25,31)(16,29,27,43)(17,30,28,22)(20,36,34,23)(21,38,35,24)(26,39,40,33)(32,42,37,41);(13,18)(14,23)(16,24)(17,28)(19,31)(20,21)(22,39)(25,36)(26,42)(27,38)(29,32)(30,33)(34,35)(37,43)(40,41);(8,10,11,9)(13,19)(18,31)(20,34)(21,35)(22,30)(29,43)(32,37)(33,39)

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, 6642);