C₇×#10303 #13187519

order	1792
nilpotency class	3
isoclinism	isoclinic class of #10300
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#10303

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₇
prime radical	C₂³×C₇	#127
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ G

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

normal subgroups only

This group has 478 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) = 39
transitive degree	μ_t(G) = 448

A minimal faithful permutation representation — 3 generators on 39 points:

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

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, 3291);