C₇×#12115 #13189331

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

Aliases

direct product

C₇×#12115

Subgroups

	subgroup	G/H
center, socle	C₂×C₇	#1595
commutator, Frattini, cosocle	C₂×M₁₆	C₂³×C₇
prime radical	C₇×#2010	C₂
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₂×C₄×C₇ ◁ C₂×C₇×M₁₆ ◁ G

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

normal subgroups only

This group has 422 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)(10,18)(11,22)(12,27)(13,17)(14,19)(15,26)(16,25)(20,38)(21,35)(23,33)(24,34)(28,31)(29,37)(30,39)(32,36);(8,10,13,21)(9,14,17,29)(11,23,24,39)(12,28,25,20)(15,31,32,38)(16,33,27,30)(18,34,35,22)(19,36,37,26);(11,24)(19,37)(23,39)(26,36)

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, 5103);