C₂₉×#1511 #16815225

order	3712
nilpotency class	3
isoclinism	isoclinic class of #1510
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₂₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂₉×#1511

Subgroups

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

274 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 92 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) = 53
transitive degree	μ_t(G) = 928

A minimal faithful permutation representation — 3 generators on 53 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)(32,36,37,34)(33,35)(38,39)(40,46,48,53)(41,45)(42,44)(43,47,51,49)(50,52);(30,32,33,37)(31,34,35,36)(38,40)(39,43)(41,49)(42,48)(44,51)(45,53)(46,52)(47,50);(39,44)(43,51)(45,52)(46,53)

Conjugacy classes & character table

754 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂₉×#8488, C₂₉×#8517, C₂₉×#8519, …
maximal normal subgroup of	C₂₉×#8488, C₂₉×#8517, C₂₉×#8519, …
maximal quotient of	C₂₉×#7593, C₂₉×#7595, C₂₉×#7600, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 243);