C₂₉×#2309 #16816023

order	3712
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄×C₂₉ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

direct product

C₂₉×#2309

Subgroups

	subgroup	G/H
center, socle	C₂³×C₂₉	C₂⁴
commutator	C₂²	C₂³×C₄×C₂₉
Frattini, cosocle	C₂³	C₂⁴×C₂₉
prime radical	C₂³×D₄×C₂₉	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₂₉ ◁ G

1104 conjugacy classes of subgroups; only the 360 normal subgroups are shown.

normal subgroups only

This group has 1104 conjugacy classes of subgroups — too many to draw. Only its 360 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 — 4 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)(31,33)(34,35,36,37)(39,42)(40,44);(38,39,41,42)(40,43,44,45);(30,31)(32,33)(34,35)(36,37)(38,39)(40,43)(41,42)(44,45);(38,40)(39,43)(41,44)(42,45)

Conjugacy classes & character table

1276 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂₉×#8748, C₂₉×#8749, C₂₉×#9497, …
maximal normal subgroup of	C₂₉×#8748, C₂₉×#8749, C₂₉×#9497, …
maximal quotient of	C₂₉×#13775, C₂₉×#13785, C₂₉×#13795, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 1041);