C₂₉×#2912 #16816626

order	3712
nilpotency class	3
isoclinism	isoclinic class of #576
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄×C₂₉ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂₉×#2912

Subgroups

	subgroup	G/H
center, socle	C₂²×C₂₉	C₂²×D₄
commutator	C₄	C₂³×C₄×C₂₉
Frattini, cosocle	C₂×C₄	C₂⁴×C₂₉
prime radical	C₂×#41×C₂₉	C₂²
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₉ ◁ C₂×#41×C₂₉ ◁ G

476 conjugacy classes of subgroups; only the 344 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 344 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) = 1856

A minimal faithful permutation representation — 4 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)(35,39)(37,44)(40,45)(41,46)(43,48)(47,49);(34,35,37,39,38,41,44,46)(36,40,43,45,42,47,48,49);(34,36)(35,40)(37,43)(38,42)(39,45)(41,47)(44,48)(46,49);(36,42)(40,47)(43,48)(45,49)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂₉×#12873, C₂₉×#12874, C₂₉×#12934, …
maximal normal subgroup of	C₂₉×#12873, C₂₉×#12874, C₂₉×#12934, …
maximal quotient of	C₂₉×#17788, C₂₉×#17798, C₂₉×#17811, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 1644);