C₂₉×#2351 #16816065

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

Aliases

direct product

C₂₉×#2351

Subgroups

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

552 conjugacy classes of subgroups; only the 368 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 368 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 — 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)(38,39,43,42)(40,45,41,44);(38,40,43,41)(39,44,42,45);(30,31,32,33)(38,41,43,40)(39,44,42,45);(38,42,43,39)(40,44,41,45)

Conjugacy classes & character table

1450 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂₉×#8766, C₂₉×#8767, C₂₉×#9176, …
maximal normal subgroup of	C₂₉×#8766, C₂₉×#8767, C₂₉×#9176, …
maximal quotient of	C₂₉×#13804, Q₈×C₂₉×#95, C₂₉×#13821, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 1083);