C₇×#12309 #13189525

order	1792
nilpotency class	4
isoclinism	isoclinic class of #12307
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#12309

Subgroups

	subgroup	G/H
center, socle	C₂×C₇	#1616
commutator, Frattini, cosocle	C₂×M₁₆	C₂³×C₇
prime radical	C₇×#469	C₂²
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₂×C₄×C₇ ◁ C₂×C₇×M₁₆ ◁ G

280 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 70 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)(8,9,14,18,13,19,12,16)(10,20,22,37,24,38,23,34)(11,25,28,35,27,33,29,17)(15,30,31,21,32,26,36,39);(8,10)(9,15)(11,26)(12,22)(13,24)(14,23)(16,31)(17,34)(18,36)(19,32)(20,33)(21,28)(25,38)(27,30)(29,39)(35,37);(9,16,19,18)(11,27)(15,31,32,36)(17,33,35,25)(20,37,38,34)(21,39)(26,30)(28,29)

Conjugacy classes & character table

175 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 5297);