C₂₉×#1996 #16815710

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

Aliases

direct product

C₂₉×#1996

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

324 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 116 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)(32,36)(33,34)(35,37)(38,39)(40,42,44,45)(41,43);(31,34)(35,36)(38,40)(39,42,43,45)(41,44);(30,32,33,37)(31,35,34,36)

Conjugacy classes & character table

928 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂₉×#8467, C₂₉×#8468, C₂₉×#8480, …
maximal normal subgroup of	C₂₉×#8467, C₂₉×#8468, C₂₉×#8480, …
maximal quotient of	C₂₉×#9878, C₂₉×#9879, C₂₉×#9880, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 728);