C₇×#12365 #13189581

order	1792
nilpotency class	4
isoclinism	isoclinic class of #12364
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#12365

Subgroups

	subgroup	G/H
center, socle	C₂×C₇	#1629
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₇
prime radical	C₇×#3291	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₂×C₄×C₇ ◁ C₄×C₇×C₈ ◁ G

346 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 78 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)(11,22,24,39)(12,17)(13,27)(14,29)(15,21)(16,20,33,38)(18,26)(23,31,30,36)(25,34)(28,35,37,32);(8,10,12,21)(9,15,17,19)(11,20,24,38)(13,14,25,26)(16,31,33,36)(18,34,29,27)(22,32,39,35)(23,28,30,37);(8,11)(9,16)(10,20)(12,24)(13,28)(14,30)(15,31)(17,33)(18,35)(19,36)(21,38)(22,34)(23,26)(25,37)(27,39)(29,32)

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, 5353);