C₇×#10011 #13187227

order	1792
nilpotency class	3
isoclinism	isoclinic class of #10004
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#10011

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₇
prime radical	C₂³×C₇	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ G

496 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 102 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)(8,9,14,19)(10,20,11,22)(12,18,13,17)(15,23,16,21)(24,25,29,33)(26,34,27,31)(28,37,38,32)(30,36,39,35);(8,10,14,11)(9,15,19,16)(12,23,13,21)(17,22,18,20)(24,26,29,27)(25,30,33,39)(28,36,38,35)(31,37,34,32);(8,11,14,10)(9,16,19,15)(12,23,13,21)(17,22,18,20)(24,27,29,26)(25,31,33,34)(28,35,38,36)(30,32,39,37)

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, 2999);