C₂₉×#1615 #16815329

order	3712
nilpotency class	3
isoclinism	isoclinic class of #1595
automorphism	?
homology	Abelianization (H₁) = C₂³×C₂₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₂₉×#1615

Subgroups

	subgroup	G/H
center, socle	C₂²×C₂₉	#120
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₂₉
prime radical	C₂₉×#121	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₉ ◁ C₂²×C₄×C₂₉ ◁ G

208 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 64 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 — 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)(30,31,37,47)(32,41,52,61)(33,35,40,36)(34,46,53,57)(38,54,60,43)(39,50,49,51)(42,58,55,48)(44,56,45,59);(30,32)(31,38)(33,42)(34,48)(35,45)(36,44)(37,52)(39,56)(40,55)(41,46)(43,51)(47,60)(49,59)(50,54)(53,58)(57,61);(30,33,36,31)(32,42,44,38)(34,49,51,57)(35,47,37,40)(39,50,46,53)(41,58,56,54)(43,61,48,59)(45,60,52,55)

Conjugacy classes & character table

667 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×C₂₉×#1615, C₂₉×#18107, C₂₉×#18127, …
maximal normal subgroup of	C₂×C₂₉×#1615, C₂₉×#18107, C₂₉×#18127, …
maximal quotient of	C₂₉×#7787, C₂₉×#7788, C₂₉×#7798, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 347);