C₂₉×#1497 #16815211

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

Aliases

direct product

C₂₉×#1497

Subgroups

	subgroup	G/H
center	C₂×C₄×C₂₉	C₂×D₄
commutator	C₂×C₄	C₂²×C₄×C₂₉
Frattini, cosocle	C₄²	C₂³×C₂₉
socle	C₂²×C₂₉	C₂×C₂²⋊C₄
prime radical	C₂³×C₂₉	C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₉ ◁ C₂×C₄²×C₂₉ ◁ G

202 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 202 conjugacy classes of subgroups — too many to draw. Only its 92 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) = 61
transitive degree	μ_t(G) = 1856

A minimal faithful permutation representation — 3 generators on 61 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)(32,39)(33,37)(34,43)(35,38)(36,41)(40,44)(42,45)(46,47,49,53,50,52,56,59)(48,51,55,57,54,58,60,61);(30,32,35,42)(31,36,38,44)(33,41,34,40)(37,45,43,39)(46,48)(47,51)(49,55)(50,54)(52,58)(53,57)(56,60)(59,61);(30,33,35,34)(31,37,38,43)(32,40,42,41)(36,39,44,45)(47,52)(48,54)(53,59)(55,60)

Conjugacy classes & character table

928 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂₉×#8490, C₂₉×#8501, C₂₉×#8502, …
maximal normal subgroup of	C₂₉×#8490, C₂₉×#8501, C₂₉×#8502, …
maximal quotient of	C₂₉×#7701, C₂₉×#7702, C₂₉×#7703, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 229);