C₂×C₇×#1710 #13196814

order	1792
nilpotency class	3
isoclinism	isoclinic class of #1705
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₇ · Schur multiplier (H₂) = C₂³×C₄

Aliases

direct product

C₂×C₇×#1710

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³×C₇	#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₇
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₂×C₄²×C₇ ◁ G

750 conjugacy classes of subgroups; only the 318 normal subgroups are shown.

normal subgroups only

This group has 750 conjugacy classes of subgroups — too many to draw. Only its 318 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) = 41
transitive degree	μ_t(G) = 1792

A minimal faithful permutation representation — 4 generators on 41 points:

(1,2)(35,36,37,38,39,40,41);(3,4,7,9)(5,11,13,18)(6,14,15,10)(8,16,17,12)(19,20)(21,27)(22,24)(23,31)(25,29)(26,30)(28,33)(32,34);(3,5,6,12,7,13,15,16)(4,8,10,11,9,17,14,18)(20,24)(25,32)(26,31)(27,33);(4,9)(8,17)(10,14)(11,18)(19,21,23,29,22,28,30,34)(20,25,26,27,24,32,31,33)

Conjugacy classes & character table

532 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 12586);