C₂×C₇×#1423 #13190847

order	1792
nilpotency class	5
isoclinism	isoclinic class of D₃₂
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂×C₇×#1423

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³×C₇	D₁₆
commutator	C₁₆	C₂²×C₄×C₇
Frattini, cosocle	C₂×C₁₆	C₂³×C₇
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₂²×C₄×C₇ ◁ C₂²×C₇×C₈ ◁ C₂²×C₇×C₁₆ ◁ G

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

normal subgroups only

This group has 206 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) = 45
transitive degree	μ_t(G) = 1792

A minimal faithful permutation representation — 3 generators on 45 points:

(1,2)(39,40,41,42,43,44,45);(3,4,5,6)(8,13)(9,14)(10,18)(11,21)(15,30)(16,33)(17,26)(19,35)(20,29)(22,34)(23,24)(25,31)(27,28)(32,38)(36,37);(7,8,14,24,10,15,27,33,21,31,35,38,34,36,20,26,12,17,29,37,22,32,19,25,11,16,28,30,18,23,9,13)

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, 6619);