C₂₉×#1776 #16815490

order	3712
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂×C₄²×C₂₉ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₂₉×#1776

Subgroups

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

272 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 168 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) = 3712

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)(30,31,33,32)(38,39,41,43,42,44,40,45);(34,35,36,37)(39,43)(40,41)(44,45);(30,32,33,31)(39,44)(43,45)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂₉×#12650, C₂₉×#12677, C₂₉×#13119, …
maximal normal subgroup of	C₂₉×#12650, C₂₉×#12677, C₂₉×#13119, …
maximal quotient of	C₂₉×#8275, C₂₉×#8276, C₂₉×#8279, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 508);