C₇×#13355 #13190571

order	1792
nilpotency class	4
isoclinism	isoclinic class of #13353
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₇×#13355

Subgroups

	subgroup	G/H
center, socle	C₂×C₇	#2021
commutator, Frattini, cosocle	C₂×#41	C₂³×C₇
prime radical	C₇×#3027	C₂
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₂×C₄×C₇ ◁ C₂×C₇×#41 ◁ G

704 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 704 conjugacy classes of subgroups — too many to draw. Only its 94 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)(8,9,13,18)(10,19,22,32)(11,23,26,16)(12,15,27,17)(14,24,29,20)(21,34,37,35)(25,33,39,31)(28,36,30,38);(8,10)(9,14)(11,24)(12,21)(13,22)(15,30)(16,32)(17,28)(18,29)(19,23)(20,26)(25,38)(27,37)(31,35)(33,34)(36,39);(9,15)(12,27)(14,28)(16,31)(17,18)(19,34)(20,24)(21,37)(23,33)(25,39)(29,30)(32,35)

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, 6343);