C₇×#9429 #13186645

order	1792
nilpotency class	3
isoclinism	isoclinic class of #9415
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₇×#9429

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#386
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

508 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 508 conjugacy classes of subgroups — too many to draw. Only its 118 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)(8,9,13,15,12,10,11,14)(16,17,21,25,22,27,20,26)(18,28,31,38,32,41,30,23)(19,33,35,24,36,39,34,40)(29,37,44,46,45,47,43,42);(8,10,12,9)(11,14,13,15)(16,18)(17,23,27,38)(19,29)(20,30)(21,31)(22,32)(24,37,40,47)(25,28,26,41)(33,42,39,46)(34,43)(35,44)(36,45);(11,13)(14,15)(16,19,22,36)(17,24,27,40)(18,29,32,45)(20,35,21,34)(23,37,38,47)(25,33,26,39)(28,42,41,46)(30,44,31,43)

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, 2417);