C₇×#12174 #13189390

order	1792
nilpotency class	4
isoclinism	isoclinic class of #12173
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#12174

Subgroups

	subgroup	G/H
center, socle	C₂×C₇	#1598
commutator, Frattini, cosocle	C₂×M₁₆	C₂³×C₇
prime radical	C₇×#575	C₂²
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₂×C₄×C₇ ◁ C₂×C₇×M₁₆ ◁ G

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

normal subgroups only

This group has 340 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)(10,20)(11,25)(12,16)(13,18)(14,19)(15,21)(17,27)(22,33)(23,36)(24,37)(26,30)(28,34)(29,35)(31,39)(32,38);(8,10,14,24)(9,15,19,32)(11,26,28,39)(12,23,13,22)(16,30,18,31)(17,33,35,36)(20,34,37,25)(21,29,38,27);(9,16,19,18)(15,30,32,31)(17,34,35,25)(20,33,37,36)

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, 5162);