C₂₉×#1820 #16815534

order	3712
nilpotency class	3
isoclinism	isoclinic class of #1462
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₂₉ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₂₉×#1820

Subgroups

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

300 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 104 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)(32,35)(33,34)(36,37)(38,39)(40,43)(41,48,49,44)(42,45)(46,52,53,50)(47,51);(31,34)(35,37)(38,40,42,47)(39,43,45,51)(41,46,49,53)(44,50,48,52);(30,32,33,36)(31,35,34,37)(38,41)(39,44)(40,46)(42,49)(43,50)(45,48)(47,53)(51,52)

Conjugacy classes & character table

928 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂₉×#7214, C₂₉×#7348, C₂₉×#8463, …
maximal normal subgroup of	C₂₉×#7214, C₂₉×#7348, C₂₉×#8463, …
maximal quotient of	C₂₉×#8699, C₂₉×#8701, C₂₉×#8703, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 552);