C₂₉×#1466 #16815180

order	3712
nilpotency class	3
isoclinism	isoclinic class of #1462
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₂₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂₉×#1466

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

300 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 100 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) = 45
transitive degree	μ_t(G) = 928

A minimal faithful permutation representation — 3 generators on 45 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)(38,39,41,43,42,44,40,45);(31,34)(32,36);(31,34)(35,37)(38,40,42,41)(39,43,44,45)

Conjugacy classes & character table

928 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂₉×#7247, C₂₉×#7320, C₂₉×#8367, …
maximal normal subgroup of	C₂₉×#7247, C₂₉×#7320, C₂₉×#8367, …
maximal quotient of	C₂₉×#7618, C₂₉×#7621, C₂₉×#7624, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 198);