C₇×#12066 #13189282

order	1792
nilpotency class	4
isoclinism	isoclinic class of #2180
automorphism	?
homology	Abelianization (H₁) = C₂²×C₇×C₈ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#12066

Subgroups

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

214 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 106 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)(8,9,11,14,12,10,13,15)(16,17,20,26,22,27,21,24)(18,28,30,45,32,46,31,42)(19,33,36,43,35,41,37,25)(23,38,39,29,40,34,44,47);(16,18,22,32)(17,23,27,40)(19,34,35,38)(20,31,21,30)(24,39,26,44)(25,42,43,45)(28,33,46,41)(29,37,47,36);(8,10,11,15,12,9,13,14)(17,24,27,26)(19,35)(23,39,40,44)(25,33,43,41)(28,42,46,45)(29,47)(34,38)(36,37)

Conjugacy classes & character table

322 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 5054);