C₂₉×#1609 #16815323

order	3712
nilpotency class	3
isoclinism	isoclinic class of #1595
automorphism	?
homology	Abelianization (H₁) = C₂³×C₂₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₂₉×#1609

Subgroups

	subgroup	G/H
center, socle	C₂²×C₂₉	#120
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₂₉
prime radical	C₂₉×#121	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₉ ◁ C₂²×C₄×C₂₉ ◁ G

216 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 64 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)(31,35)(32,33)(36,37)(38,39,42,45)(40,43,47,50)(41,44,48,51)(46,52,53,49);(30,31)(32,35)(33,37)(34,36)(38,40)(39,43)(41,46)(42,47)(44,49)(45,50)(48,53)(51,52);(30,32,34,33)(31,35,36,37)(38,41)(39,44)(40,46)(42,48)(43,49)(45,51)(47,53)(50,52)

Conjugacy classes & character table

667 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×C₂₉×#1609, C₂₉×#18107, C₂₉×#18126, …
maximal normal subgroup of	C₂×C₂₉×#1609, C₂₉×#18107, C₂₉×#18126, …
maximal quotient of	C₂₉×#7788, C₂₉×#7789, C₂₉×#7792, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 341);