C₂₉×#3166 #16816880

order	3712
nilpotency class	3
isoclinism	isoclinic class of #3055
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₂₉ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₂₉×#3166

Subgroups

	subgroup	G/H
center, socle	C₂²×C₂₉	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₂₉
prime radical	C₂³×C₂₉	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₉ ◁ C₂×C₄²×C₂₉ ◁ G

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

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 216 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) = 1856

A minimal faithful permutation representation — 4 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)(30,31,33,35)(32,36,37,34)(38,39,41,43)(40,46,47,53)(42,50,49,45)(44,52,51,48);(30,32,33,37)(31,34,35,36)(39,43)(44,51)(45,50)(46,53);(38,40,42,48,41,47,49,52)(39,44,45,46,43,51,50,53);(40,47)(44,51)(46,53)(48,52)

Conjugacy classes & character table

928 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂₉×#9440, C₂₉×#10023, C₂₉×#10025, …
maximal normal subgroup of	C₂₉×#9440, C₂₉×#10023, C₂₉×#10025, …
maximal quotient of	C₂₉×#20801, C₂₉×#21159, C₂₉×#21323, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 1898);