C₂₉×#1675 #16815389

order	3712
nilpotency class	3
isoclinism	isoclinic class of #1665
automorphism	?
homology	Abelianization (H₁) = C₂³×C₂₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₂₉×#1675

Subgroups

	subgroup	G/H
center, socle	C₂²×C₂₉	#121
commutator, Frattini, cosocle	C₄²	C₂³×C₂₉
prime radical	C₂×D₄×C₂₉	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₉ ◁ C₄²×C₂₉ ◁ G

160 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 72 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)(30,31,33,36)(32,34,37,35)(38,39,42,45)(40,46,49,53)(41,50,47,44)(43,51,52,48);(38,40,47,51,42,49,41,48)(39,43,50,53,45,52,44,46);(31,34)(32,37)(35,36)

Conjugacy classes & character table

841 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂₉×#12372, C₂₉×#12373, C₂₉×#12374, …
maximal normal subgroup of	C₂₉×#12372, C₂₉×#12373, C₂₉×#12374, …
maximal quotient of	C₂₉×#8644, C₂₉×#9215, C₂₉×#9567, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 407);