C₂₉×#1468 #16815182

order	3712
nilpotency class	3
isoclinism	isoclinic class of #1468
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₂₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂₉×#1468

Subgroups

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

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

normal subgroups only

This group has 280 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)(38,39,41,44,42,45,48,51)(40,43,47,50,46,49,52,53);(30,32)(31,34)(33,37)(35,36)(38,40)(39,43)(41,47)(42,46)(44,50)(45,49)(48,52)(51,53);(40,46)(43,49)(47,52)(50,53)

Conjugacy classes & character table

928 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂₉×#8487, C₂₉×#8488, C₂₉×#8539, …
maximal normal subgroup of	C₂₉×#8487, C₂₉×#8488, C₂₉×#8539, …
maximal quotient of	C₂₉×#7654, C₂₉×#7655, C₂₉×#7658, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 200);