C₇×#12088 #13189304

order	1792
nilpotency class	4
isoclinism	isoclinic class of #12087
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₇×#12088

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂²×C₇	#442
commutator	C₂×C₈	C₂²×C₄×C₇
Frattini, cosocle	C₄×C₈	C₂³×C₇
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₄²×C₇ ◁ C₇×#322 ◁ G

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

normal subgroups only

This group has 198 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) = 47
transitive degree	μ_t(G) = 1792

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

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

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, 5076);