C₂₉×#1519 #16815233

order	3712
nilpotency class	3
isoclinism	isoclinic class of #1504
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₂₉ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₂₉×#1519

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	C₂×C₂₉×#121	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₉ ◁ C₂³×C₄×C₂₉ ◁ G

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

normal subgroups only

This group has 340 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) = 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)(38,39)(40,44)(41,43)(42,45);(30,32)(31,34)(33,37)(35,36)(39,43)(40,41)(44,45);(32,37)(34,36)(38,40,42,41)(39,43,45,44)

Conjugacy classes & character table

928 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂₉×#8350, C₂₉×#8352, C₂₉×#8375, …
maximal normal subgroup of	C₂₉×#8350, C₂₉×#8352, C₂₉×#8375, …
maximal quotient of	C₂₉×#7572, C₂₉×#7573, C₂₉×#7586, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 251);