C₂₉×#2619 #16816333

order	3712
nilpotency class	2
isoclinism	isoclinic class of #2613
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₂₉ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

direct product

C₂₉×#2619

Subgroups

	subgroup	G/H
center, socle	C₂³×C₂₉	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₂₉
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₉ ◁ G

728 conjugacy classes of subgroups; only the 216 normal subgroups are shown.

normal subgroups only

This group has 728 conjugacy classes of subgroups — too many to draw. Only its 216 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 — 4 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)(34,35,36,37)(38,39)(40,43)(41,42)(44,45);(31,33)(39,42)(43,44);(30,31)(32,33)(34,35)(36,37)(38,39)(40,44)(41,42)(43,45);(38,40,41,45)(39,43,42,44)

Conjugacy classes & character table

928 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂₉×#9197, C₂₉×#9198, C₂₉×#11693, …
maximal normal subgroup of	C₂₉×#9197, C₂₉×#9198, C₂₉×#11693, …
maximal quotient of	C₂₉×#14257, C₂₉×#14260, C₂₉×#14272, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 1351);