C₂₉×#1515 #16815229

order	3712
nilpotency class	3
isoclinism	isoclinic class of #1510
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₂₉ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₂₉×#1515

Subgroups

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

208 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 88 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)(31,36,38,50)(32,41)(33,45)(37,52,53,61)(39,51,40,46)(43,58)(44,57)(47,48)(49,54,60,55)(56,59);(30,31,34,39)(32,37,44,54)(33,46,45,50)(35,40,42,38)(36,48,51,47)(41,55,57,53)(43,52,59,60)(49,58,61,56);(30,32,42,57)(31,37,38,53)(33,43,48,58)(34,44,35,41)(36,49,50,60)(39,54,40,55)(45,59,47,56)(46,52,51,61)

Conjugacy classes & character table

754 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂₉×#8501, C₂₉×#8502, C₂₉×#8503, …
maximal normal subgroup of	C₂₉×#8501, C₂₉×#8502, C₂₉×#8503, …
maximal quotient of	C₂₉×#7596, C₂₉×#7604, C₂₉×#7605, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 247);