C₇×#12116 #13189332

order	1792
nilpotency class	4
isoclinism	isoclinic class of #12114
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#12116

Subgroups

	subgroup	G/H
center, socle	C₂×C₇	#1595
commutator, Frattini, cosocle	C₂×M₁₆	C₂³×C₇
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₂×C₄×C₇ ◁ C₂×C₇×M₁₆ ◁ G

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

normal subgroups only

This group has 426 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)(10,19)(11,16)(12,24)(13,28)(14,18)(15,21)(17,26)(20,35)(22,36)(23,34)(25,30)(27,39)(29,33)(31,38)(32,37);(8,10)(9,15)(11,20)(12,25)(13,29)(14,22)(16,30)(17,33)(18,31)(19,24)(21,26)(23,38)(27,37)(28,32)(34,35)(36,39);(8,11,14,13)(9,16,18,28)(10,20,22,29)(12,26,27,23)(15,30,31,32)(17,24,34,39)(19,35,36,33)(21,37,38,25)

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, 5104);