C₂×C₇×#1358 #13189135

order	1792
nilpotency class	4
isoclinism	isoclinic class of #1357
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂×C₇×#1358

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#331
commutator	C₄²	C₂²×C₄×C₇
Frattini, cosocle	C₄×C₈	C₂³×C₇
prime radical	C₂³×C₇	#100
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂²×C₄×C₇ ◁ C₂×C₄×C₇×C₈ ◁ G

390 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 142 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) = 448

A minimal faithful permutation representation — 3 generators on 41 points:

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

Conjugacy classes & character table

322 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 4907);