C₇×#12079 #13189295

order	1792
nilpotency class	4
isoclinism	isoclinic class of #12077
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₇×#12079

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

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

normal subgroups only

This group has 204 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)(8,9,11,10,12,13,15,14)(16,17,20,24,22,26,21,37)(18,27,29,45,31,46,30,42)(19,32,35,44,34,43,36,25)(23,38,40,28,41,33,39,47);(16,18,22,31)(17,23,26,41)(19,33,34,38)(20,30,21,29)(24,39,37,40)(25,42,44,45)(27,32,46,43)(28,36,47,35);(8,10,11,13,12,14,15,9)(17,24,26,37)(19,34)(23,39,41,40)(25,43,44,32)(27,45,46,42)(28,47)(33,38)(35,36)

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, 5067);