C₂₉×#2140 #16815854

order	3712
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₂₉ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₂₉×#2140

Subgroups

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

260 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 104 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) = 928

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)(36,41)(37,40)(38,47)(39,43)(42,45)(44,48)(46,49);(35,40)(36,38)(37,45)(41,42)(43,49)(44,46)(47,48);(34,36,39,38)(35,41,43,47)(37,44,46,45)(40,48,49,42)

Conjugacy classes & character table

928 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂₉×#12647, C₂₉×#12741, C₂₉×#12750, …
maximal normal subgroup of	C₂₉×#12647, C₂₉×#12741, C₂₉×#12750, …
maximal quotient of	C₂₉×#11593, C₂₉×#11649, C₂₉×#11661, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 872);