C₂₉×#2226 #16815940

order	3712
nilpotency class	4
isoclinism	isoclinic class of #2184
automorphism	?
homology	Abelianization (H₁) = C₂³×C₂₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₂₉×#2226

Subgroups

	subgroup	G/H
center, socle	C₂²×C₂₉	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂³×C₂₉
prime radical	C₂×D₈×C₂₉	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₉ ◁ C₄²×C₂₉ ◁ C₄×C₈×C₂₉ ◁ G

142 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 142 conjugacy classes of subgroups — too many to draw. Only its 68 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 — 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)(30,31,33,35)(32,36,37,34)(38,39,42,46)(40,43,49,52)(41,50,48,45)(44,53,51,47);(30,32,33,37)(31,34,35,36);(39,43)(40,47)(41,48)(44,45)(46,52)(49,53)(50,51)

Conjugacy classes & character table

841 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂₉×#12372, C₂₉×#12375, C₂₉×#12378, …
maximal normal subgroup of	C₂₉×#12372, C₂₉×#12375, C₂₉×#12378, …
maximal quotient of	C₂₉×#12811, C₂₉×#12937, C₂₉×#13111, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 958);