C₇×#10676 #13187892

order	1792
nilpotency class	3
isoclinism	isoclinic class of #10304
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₇×#10676

Subgroups

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

410 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 162 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) = 1792

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

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

Conjugacy classes & character table

364 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 3664);