C₂₉×#1469 #16815183

order	3712
nilpotency class	3
isoclinism	isoclinic class of #1468
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₂₉ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₂₉×#1469

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 ◁ C₂×C₄²×C₂₉ ◁ G

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

normal subgroups only

This group has 256 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)(30,31,33,37,34,35,32,36)(38,39)(40,46,48,43)(41,44)(42,45)(47,52,53,50)(49,51);(31,35)(32,33)(38,40)(39,43)(41,47)(42,48)(44,50)(45,46)(49,53)(51,52);(30,32,34,33)(31,36,35,37)(38,41,42,49)(39,44,45,51)(40,47,48,53)(43,50,46,52)

Conjugacy classes & character table

928 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂₉×#8475, C₂₉×#8477, C₂₉×#8483, …
maximal normal subgroup of	C₂₉×#8475, C₂₉×#8477, C₂₉×#8483, …
maximal quotient of	C₂₉×#7656, C₂₉×#7659, C₂₉×#7662, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 201);