C₂₉×#1504 #16815218

order	3712
nilpotency class	3
isoclinism	isoclinic class of #1504
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₂₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂₉×#1504

Subgroups

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

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

normal subgroups only

This group has 334 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) = 464

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

Conjugacy classes & character table

928 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂₉×#8346, C₂₉×#8347, C₂₉×#8453, …
maximal normal subgroup of	C₂₉×#8346, C₂₉×#8347, C₂₉×#8453, …
maximal quotient of	C₂₉×#7578, C₂₉×#7579, C₂₉×#7586, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 236);