C₂×C₇×#1413 #13190813

order	1792
nilpotency class	5
isoclinism	isoclinic class of #1412
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂×C₇×#1413

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#354
commutator	C₄²	C₂²×C₄×C₇
Frattini, cosocle	C₄⋊Q₈	C₂³×C₇
prime radical	C₂³×C₇	#99
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂³×C₇ ◁ C₂²×C₄×C₇ ◁ C₂×C₇×C₄⋊Q₈ ◁ G

314 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 94 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);(3,4,6,11,8,13,20,30)(5,14,16,32,15,31,33,28)(7,12,22,29,23,10,19,27)(9,24,25,17,26,18,34,21);(3,5,8,15)(4,9)(6,18,20,24)(7,21,23,17)(10,14)(11,28)(12,31)(13,26)(16,19,33,22)(25,27)(29,34)(30,32)

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 6585);