C₇×#13429 #13190645

order	1792
nilpotency class	4
isoclinism	isoclinic class of #13429
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#13429

Subgroups

	subgroup	G/H
center, socle	C₂×C₇	#2044
commutator, Frattini, cosocle	C₂×#41	C₂³×C₇
prime radical	C₇×#3199	C₂
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₂×C₄×C₇ ◁ C₂×C₇×#41 ◁ G

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

normal subgroups only

This group has 434 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) = 39
transitive degree	μ_t(G) = 224

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

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

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 6417);