C₂₉×#1554 #16815268

order	3712
nilpotency class	3
isoclinism	isoclinic class of #1504
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₂₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₂₉×#1554

Subgroups

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

198 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 92 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) = 1856

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

Conjugacy classes & character table

928 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂₉×#8490, C₂₉×#8501, C₂₉×#8503, …
maximal normal subgroup of	C₂₉×#8490, C₂₉×#8501, C₂₉×#8503, …
maximal quotient of	C₂₉×#7581, C₂₉×#7583, C₂₉×#7611, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 286);