C₂₉×#3414 #16817128

order	3712
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₂₉ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂₉×#3414

Subgroups

	subgroup	G/H
center	C₄×C₂₉	C₂×D₈
commutator, Frattini, cosocle	C₈	C₂⁴×C₂₉
socle	C₂×C₂₉	C₂²×D₈
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₂₉ ◁ C₂²×C₄×C₂₉ ◁ C₂²×C₈×C₂₉ ◁ G

364 conjugacy classes of subgroups; only the 180 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 180 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) = 928

A minimal faithful permutation representation — 4 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)(32,41)(33,39)(34,46)(35,50)(36,38)(37,43)(40,48)(42,58)(44,57)(45,55)(47,53)(49,60)(51,56)(52,59)(54,61);(30,32)(31,37)(33,42)(34,47)(35,51)(36,44)(38,52)(39,53)(40,56)(41,46)(43,48)(45,59)(49,61)(50,54)(55,58)(57,60);(31,38)(37,52)(39,50)(40,55)(41,57)(46,60)(53,54)(56,58);(30,33,36,35)(31,39,38,50)(32,42,44,51)(34,45,49,48)(37,53,52,54)(40,46,55,60)(41,58,57,56)(43,47,59,61)

Conjugacy classes & character table

928 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂₉×#13021, C₂₉×#13024, C₂₉×#13027, …
maximal normal subgroup of	C₂₉×#13021, C₂₉×#13024, C₂₉×#13027, …
maximal quotient of	C₂₉×#33380, C₂₉×#33409, C₂₉×#33451, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 2146);