C₂₉×#1836 #16815550

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

Aliases

direct product

C₂₉×#1836

Subgroups

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

224 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 120 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) = 49
transitive degree	μ_t(G) = 3712

A minimal faithful permutation representation — 3 generators on 49 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)(34,35)(36,39)(37,38)(40,41)(42,43,47,44)(45,46,49,48);(35,38)(39,41)(42,44,46,49,47,43,48,45);(34,36,37,40)(35,39,38,41)(42,45,46,44,47,49,48,43)

Conjugacy classes & character table

928 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂₉×#7266, C₂₉×#7367, C₂₉×#8911, …
maximal normal subgroup of	C₂₉×#7266, C₂₉×#7367, C₂₉×#8911, …
maximal quotient of	C₂₉×#8786, C₂₉×#8789, C₂₉×#8794, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 568);