C₇×#8794 #13186010

order	1792
nilpotency class	3
isoclinism	isoclinic class of #8794
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₈

Aliases

direct product

C₇×#8794

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#382
commutator	C₄²	C₂²×C₄×C₇
Frattini, cosocle	C₂×C₄²	C₂³×C₇
prime radical	C₂³×C₇	#124
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ G

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

normal subgroups only

This group has 270 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) = 47
transitive degree	μ_t(G) = 448

A minimal faithful permutation representation — 3 generators on 47 points:

(1,2,3,4,5,6,7)(8,9,13,10)(11,12,15,14)(16,17)(18,26,29,22)(19,30,33,45)(20,25)(21,24)(23,32,38,42)(27,41)(28,40,44,36)(31,46)(34,39)(35,43)(37,47);(8,10,12,15,13,9,14,11)(16,18,20,28,21,29,34,44)(17,22,25,36,24,26,39,40)(19,31,32,47,33,27,42,43)(23,37,30,41,38,35,45,46);(8,11,12,10,13,15,14,9)(16,19)(17,23,24,38)(18,27)(20,32)(21,33)(22,35,26,37)(25,30,39,45)(28,43)(29,31)(34,42)(36,46,40,41)(44,47)

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, 1782);