C₂₉×#1501 #16815215

order	3712
nilpotency class	3
isoclinism	isoclinic class of #327
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₂₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂₉×#1501

Subgroups

	subgroup	G/H
center	C₂×C₄×C₂₉	C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₄×C₂₉
Frattini, cosocle	C₂²×C₄	C₂³×C₂₉
socle	C₂²×C₂₉	C₂×C₂²⋊C₄
prime radical	C₂₉×#525	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₉ ◁ C₂³×C₄×C₂₉ ◁ G

304 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 96 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)(32,36,37,34)(33,35)(39,43,44,45);(30,32)(31,34)(33,37)(35,36)(38,39)(40,43)(41,45)(42,44);(38,40,42,41)(39,43,44,45)

Conjugacy classes & character table

1102 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂₉×#8238, C₂₉×#8239, C₂₉×#8240, …
maximal normal subgroup of	C₂₉×#8238, C₂₉×#8239, C₂₉×#8240, …
maximal quotient of	C₂₉×#7563, C₂₉×#7564, C₂₉×#7565, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 233);