C₇×#12117 #13189333

order	1792
nilpotency class	4
isoclinism	isoclinic class of #12117
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#12117

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

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

normal subgroups only

This group has 424 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)(10,21)(11,15)(12,19)(13,18)(14,20)(16,23)(17,25)(22,34)(24,35)(26,31)(27,33)(28,30)(29,38)(32,39)(36,37);(9,15)(11,25)(12,13)(16,29)(17,19)(18,30)(20,31)(21,34)(22,37)(23,27)(26,28)(32,35)(33,36)(38,39);(8,10)(9,16)(11,22)(12,27)(13,23)(14,24)(15,29)(17,34)(18,35)(19,21)(20,33)(25,37)(26,39)(28,38)(30,32)(31,36)

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, 5105);