C₂₉×#1995 #16815709

order	3712
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₂₉ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₂₉×#1995

Subgroups

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

244 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 112 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) = 53
transitive degree	μ_t(G) = 928

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

Conjugacy classes & character table

928 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂₉×#7129, C₂₉×#7360, C₂₉×#8494, …
maximal normal subgroup of	C₂₉×#7129, C₂₉×#7360, C₂₉×#8494, …
maximal quotient of	C₂₉×#9872, C₂₉×#9873, C₂₉×#9874, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 727);