C₇×#12458 #13189674

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

Aliases

direct product

C₇×#12458

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

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

normal subgroups only

This group has 192 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) = 55
transitive degree	μ_t(G) = 1792

A minimal faithful permutation representation — 3 generators on 55 points:

(1,2,3,4,5,6,7)(8,9,12,15)(10,16,18,23)(11,19,21,14)(13,20,22,17)(24,25,29,33)(26,34,38,50)(27,39,41,54)(28,42,40,32)(30,43,45,36)(31,46,47,35)(37,53,52,49)(44,51,48,55);(24,26,35,51,28,37,41,43,29,38,46,55,40,52,27,36)(25,30,39,49,32,44,47,50,33,45,54,53,42,48,31,34);(8,10,12,18)(9,13,15,22)(11,20,21,17)(14,23,19,16)

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, 5446);