C₂₉×#2011 #16815725

order	3712
nilpotency class	3
isoclinism	isoclinic class of #1999
automorphism	?
homology	Abelianization (H₁) = C₂³×C₂₉ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂₉×#2011

Subgroups

	subgroup	G/H
center, socle	C₂³×C₂₉	C₂×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₂₉
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₉ ◁ C₂×C₄²×C₂₉ ◁ G

510 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 510 conjugacy classes of subgroups — too many to draw. Only its 116 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) = 1856

A minimal faithful permutation representation — 3 generators on 45 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)(38,39)(40,43)(41,44)(42,45);(31,33)(34,35)(36,37)(39,43)(40,41)(44,45);(35,37)(39,44)(40,41)(43,45)

Conjugacy classes & character table

928 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂₉×#7402, C₂₉×#7734, C₂₉×#7735, …
maximal normal subgroup of	C₂₉×#7402, C₂₉×#7734, C₂₉×#7735, …
maximal quotient of	C₂₉×#9946, C₂₉×#9952, C₂₉×#9962, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 743);