C₂₉×#1961 #16815675

order	3712
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₂₉ · Schur multiplier (H₂) = C₂³×C₄

Aliases

direct product

C₂₉×#1961

Subgroups

	subgroup	G/H
center, socle	C₂³×C₂₉	C₂×D₄
commutator	C₂×C₄	C₂²×C₄×C₂₉
Frattini, cosocle	C₂²×C₄	C₂³×C₂₉
prime radical	C₂×C₂₉×#127	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₉ ◁ C₂×C₄²×C₂₉ ◁ G

412 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 128 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)(31,33)(34,35,36,37)(38,39)(40,44)(41,45)(42,43);(35,37)(38,40,44,39,42,45,41,43);(30,31,32,33)(39,43)(40,45)

Conjugacy classes & character table

928 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂₉×#7116, C₂₉×#7352, C₂₉×#7504, …
maximal normal subgroup of	C₂₉×#7116, C₂₉×#7352, C₂₉×#7504, …
maximal quotient of	C₂₉×#9642, C₂₉×#9648, C₂₉×#9684, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 693);