C₂₉×#2316 #16816030

order	3712
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄×C₂₉ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂₉×#2316

Subgroups

	subgroup	G/H
center, socle	C₂³×C₂₉	C₂⁴
commutator	C₂²	C₂³×C₄×C₂₉
Frattini, cosocle	C₂³	C₂⁴×C₂₉
prime radical	C₂⁴×C₂₉	C₂³
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₂₉ ◁ G

600 conjugacy classes of subgroups; only the 360 normal subgroups are shown.

normal subgroups only

This group has 600 conjugacy classes of subgroups — too many to draw. Only its 360 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)(31,33)(34,35,36,37)(38,39)(40,44)(41,43)(42,45);(38,39,41,43)(40,42,45,44);(30,31,32,33)(38,39)(40,42)(41,43)(44,45);(38,40)(39,42)(41,45)(43,44)

Conjugacy classes & character table

1276 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂₉×#8757, C₂₉×#8856, C₂₉×#8857, …
maximal normal subgroup of	C₂₉×#8757, C₂₉×#8856, C₂₉×#8857, …
maximal quotient of	C₂₉×#13778, C₂₉×#13783, C₂₉×#13795, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 1048);