C₂₉×#2159 #16815873

order	3712
nilpotency class	4
isoclinism	isoclinic class of #509
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₂₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₂₉×#2159

Subgroups

	subgroup	G/H
center, socle	C₂²×C₂₉	C₂×D₈
commutator	C₈	C₂²×C₄×C₂₉
Frattini, cosocle	C₂×C₈	C₂³×C₂₉
prime radical	C₂³×C₂₉	D₈
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₉ ◁ C₂²×C₄×C₂₉ ◁ C₂²×C₈×C₂₉ ◁ G

156 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 156 conjugacy classes of subgroups — too many to draw. Only its 104 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) = 49
transitive degree	μ_t(G) = 1856

A minimal faithful permutation representation — 3 generators on 49 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29)(30,31,32,33)(35,39)(36,40)(37,43)(41,45)(42,49)(44,48)(46,47);(34,35,40,47,37,42,44,45,38,41,48,49,43,46,36,39);(35,41)(39,45)(42,46)(47,49)

Conjugacy classes & character table

928 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂₉×#12677, C₂₉×#12923, C₂₉×#12925, …
maximal normal subgroup of	C₂₉×#12677, C₂₉×#12923, C₂₉×#12925, …
maximal quotient of	C₂₉×#12009, C₂₉×#12025, C₂₉×#12035, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 891);