C₇×#12367 #13189583

order	1792
nilpotency class	4
isoclinism	isoclinic class of #12366
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#12367

Subgroups

	subgroup	G/H
center, socle	C₂×C₇	#1629
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₇
prime radical	C₇×#3293	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₂×C₄×C₇ ◁ C₄×C₇×C₈ ◁ G

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

normal subgroups only

This group has 336 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)(10,18,20,37)(11,22)(12,25)(14,29,31,39)(15,33)(16,28)(19,24,21,36)(23,35)(26,27)(30,34,32,38);(8,9,13,17)(10,14,20,31)(11,23,27,33)(12,28,25,16)(15,22,35,26)(18,34,37,38)(19,32,21,30)(24,29,36,39);(8,10)(9,14)(11,24)(12,19)(13,20)(15,34)(16,30)(17,31)(18,26)(21,25)(22,37)(23,29)(27,36)(28,32)(33,39)(35,38)

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, 5355);