C₇×#12440 #13189656

order	1792
nilpotency class	4
isoclinism	isoclinic class of #12430
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₇×#12440

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂²×C₇	#466
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₇
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₄²×C₇ ◁ C₇×C₈×Q₈ ◁ G

216 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 86 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)(9,12)(10,14)(13,15)(16,17,21,25)(18,26,30,42)(19,31,33,46)(20,34,32,24)(22,35,37,28)(23,38,39,27)(29,45,44,41)(36,43,40,47);(16,18,27,43,20,29,33,35,21,30,38,47,32,44,19,28)(17,22,31,41,24,36,39,42,25,37,46,45,34,40,23,26);(8,9,11,13)(10,15,14,12)

Conjugacy classes & character table

343 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 5428);