C₂₉×#1991 #16815705

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

Aliases

direct product

C₂₉×#1991

Subgroups

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

252 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 128 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) = 57
transitive degree	μ_t(G) = 3712

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

Conjugacy classes & character table

1276 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂₉×#7135, C₂₉×#7359, C₂₉×#8529, …
maximal normal subgroup of	C₂₉×#7135, C₂₉×#7359, C₂₉×#8529, …
maximal quotient of	C₂₉×#9871, C₂₉×#9875, C₂₉×#9886, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 723);