C₂₉×#3416 #16817130

order	3712
nilpotency class	4
isoclinism	isoclinic class of #3415
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₂₉ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂₉×#3416

Subgroups

	subgroup	G/H
center, socle	C₂×C₂₉	C₂²×D₈
commutator, Frattini, cosocle	C₈	C₂⁴×C₂₉
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₂₉ ◁ #41×C₂₉ ◁ C₂₉×#131 ◁ G

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

normal subgroups only

This group has 360 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)(30,31,35,40)(32,37,41,51)(33,36,46,50)(34,47,45,39)(38,54,49,44)(42,48,55,59)(43,58,56,53)(52,61,60,57);(31,36)(33,44)(34,45)(37,48)(38,39)(40,50)(42,57)(43,56)(46,54)(47,49)(51,59)(52,53)(55,61)(58,60);(30,32)(31,37)(33,42)(34,43)(35,41)(36,48)(38,52)(39,53)(40,51)(44,57)(45,56)(46,55)(47,58)(49,60)(50,59)(54,61);(32,41)(37,51)(42,55)(43,56)(48,59)(52,60)(53,58)(57,61)

Conjugacy classes & character table

928 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂₉×#33854, C₂₉×#33855, C₂₉×#33856, …
maximal normal subgroup of	C₂₉×#33854, C₂₉×#33855, C₂₉×#33856, …
maximal quotient of	C₂₉×#33381, C₂₉×#33382, C₂₉×#33411, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 2148);