C₇×#12438 #13189654

order	1792
nilpotency class	4
isoclinism	isoclinic class of #12436
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#12438

Subgroups

	subgroup	G/H
center, socle	C₂×C₇	#1667
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₇
prime radical	C₇×#2213	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₂×C₄×C₇ ◁ C₄×C₇×C₈ ◁ G

224 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 74 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)(9,14)(10,19)(11,24)(12,25)(15,26)(17,31)(18,30)(20,21)(22,37)(23,32)(27,33)(28,29)(34,35)(38,39);(8,9,15,31,12,17,11,14,13,18,24,29,25,28,26,30)(10,16,32,38,21,34,20,27,23,36,19,33,22,35,37,39);(8,10)(9,16)(11,20)(12,21)(13,23)(14,27)(15,32)(17,34)(18,36)(19,24)(22,25)(26,37)(28,35)(29,33)(30,39)(31,38)

Conjugacy classes & character table

259 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 5426);