C₂₉×#1819 #16815533

order	3712
nilpotency class	3
isoclinism	isoclinic class of #1462
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₂₉ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₂₉×#1819

Subgroups

	subgroup	G/H
center, socle	C₂²×C₂₉	C₂×C₂²⋊C₄
commutator	C₂³	C₂²×C₄×C₂₉
Frattini, cosocle	C₂⁴	C₂³×C₂₉
prime radical	C₂⁵×C₂₉	C₂²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₉ ◁ C₂⁵×C₂₉ ◁ G

504 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 112 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) = 464

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)(30,31)(32,38,40,36)(33,35)(34,37)(39,44,45,42)(41,43);(31,35)(36,42)(37,43)(38,44);(30,32,33,39)(31,36,35,42)(34,40,41,45)(37,38,43,44)

Conjugacy classes & character table

928 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂₉×#7213, C₂₉×#7349, C₂₉×#8535, …
maximal normal subgroup of	C₂₉×#7213, C₂₉×#7349, C₂₉×#8535, …
maximal quotient of	C₂₉×#8698, C₂₉×#8700, C₂₉×#8702, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 551);