C₂₉×#1752 #16815466

order	3712
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂×C₄²×C₂₉ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₂₉×#1752

Subgroups

	subgroup	G/H
center	C₂²×C₄×C₂₉	C₂³
commutator	C₂²	C₂×C₄²×C₂₉
Frattini, cosocle	C₂²×C₄	C₂³×C₂₉
socle	C₂³×C₂₉	C₂²×C₄
prime radical	C₂⁴×C₂₉	C₂×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₂₉ ◁ G

360 conjugacy classes of subgroups; only the 184 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 184 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)(38,39,42,44,41,40,43,45);(31,33)(34,35,36,37)(39,40)(44,45);(38,40,42,45,41,39,43,44)

Conjugacy classes & character table

1624 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂₉×#7200, C₂₉×#7201, C₂₉×#8160, …
maximal normal subgroup of	C₂₉×#7200, C₂₉×#7201, C₂₉×#8160, …
maximal quotient of	C₂₉×#8144, C₂₉×#8145, C₂₉×#8149, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 484);