C₂₉×#1580 #16815294

order	3712
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈×C₂₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂₉×#1580

Subgroups

	subgroup	G/H
center	C₂×C₄×C₂₉	C₂×D₄
commutator	C₄	C₂²×C₈×C₂₉
Frattini, cosocle	C₄²	C₂³×C₂₉
socle	C₂²×C₂₉	C₄×D₄
prime radical	C₂×D₈×C₂₉	C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₉ ◁ C₄×C₈×C₂₉ ◁ G

192 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 100 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)(30,31,33,35)(32,34,37,36)(38,39,41,44,42,40,43,45);(31,34)(32,37)(35,36);(31,35)(34,36)(38,40,41,45,42,39,43,44)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂₉×#12065, C₂₉×#12067, C₂₉×#12077, …
maximal normal subgroup of	C₂₉×#12065, C₂₉×#12067, C₂₉×#12077, …
maximal quotient of	C₂₉×#8200, C₂₉×#8284, C₂₉×#8845, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 312);