C₇×#8492 #13185708

order	1792
nilpotency class	3
isoclinism	isoclinic class of #8491
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₇×#8492

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#379
commutator	C₄²	C₂²×C₄×C₇
Frattini, cosocle	C₂×C₄²	C₂³×C₇
prime radical	C₇×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ G

692 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 692 conjugacy classes of subgroups — too many to draw. Only its 142 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)(9,12,13,14)(16,17)(18,26,29,22)(19,30,32,39)(20,25)(21,24)(23,31,37,33)(27,42)(28,41,46,36)(34,38)(35,43)(40,45)(44,47);(8,9)(10,14)(11,13)(12,15)(16,18,21,29)(17,22,24,26)(19,27,32,44)(20,28,34,46)(23,35,37,40)(25,36,38,41)(30,47,39,42)(31,43,33,45);(10,15)(12,14)(16,19)(17,23)(18,27)(20,33)(21,32)(22,35)(24,37)(25,39)(26,40)(28,45)(29,44)(30,38)(31,34)(36,42)(41,47)(43,46)

Conjugacy classes & character table

238 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 1480);