C₂₉×#1550 #16815264

order	3712
nilpotency class	3
isoclinism	isoclinic class of #1504
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₂₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₂₉×#1550

Subgroups

	subgroup	G/H
center, socle	C₂²×C₂₉	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₄×C₂₉
Frattini, cosocle	C₄²	C₂³×C₂₉
prime radical	C₂³×C₂₉	C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₉ ◁ C₂×C₄²×C₂₉ ◁ G

216 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 108 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) = 1856

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)(31,35)(32,33)(36,37)(38,39,42,45,43,40,41,44);(30,31,34,36)(32,35,33,37)(38,40,43,39)(41,44,42,45);(30,32,34,33)(31,35,36,37)

Conjugacy classes & character table

928 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂₉×#8355, C₂₉×#8356, C₂₉×#8357, …
maximal normal subgroup of	C₂₉×#8355, C₂₉×#8356, C₂₉×#8357, …
maximal quotient of	C₂₉×#7577, C₂₉×#7589, C₂₉×#7615, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 282);