C₂₉×#2275 #16815989

order	3712
nilpotency class	2
isoclinism	isoclinic class of H(5,𝔽₂)
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄²×C₂₉ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

direct product

C₂₉×#2275

Subgroups

	subgroup	G/H
center, socle	C₂³×C₂₉	C₂⁴
commutator	C₂	C₂²×C₄²×C₂₉
Frattini, cosocle	C₂³	C₂⁴×C₂₉
prime radical	C₂²×D₄×C₂₉	C₂²
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₉ ◁ G

780 conjugacy classes of subgroups; only the 520 normal subgroups are shown.

normal subgroups only

This group has 780 conjugacy classes of subgroups — too many to draw. Only its 520 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 — 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)(30,31,33,32)(34,35,37,36)(38,39,42,40)(41,44,45,43);(34,35,37,36)(39,40)(43,44);(30,32,33,31)(34,36,37,35)(38,40,42,39)(41,44,45,43);(38,41)(39,43)(40,44)(42,45)

Conjugacy classes & character table

1972 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂₉×#9323, C₂₉×#9325, C₂₉×#17547, …
maximal normal subgroup of	C₂₉×#9323, C₂₉×#9325, C₂₉×#17547, …
maximal quotient of	C₂₉×#13749, C₂₉×#13750, C₂₉×#13782, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 1007);