C₂₉×#1535 #16815249

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

Aliases

direct product

C₂₉×#1535

Subgroups

	subgroup	G/H
center, socle	C₂²×C₂₉	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₄×C₂₉
Frattini, cosocle	C₄²	C₂³×C₂₉
prime radical	C₂₉×#121	C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₉ ◁ C₂×C₄²×C₂₉ ◁ G

200 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 84 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)(30,31,34,39,35,38,33,37)(32,40,41,45,42,36,43,44)(46,47)(48,54,56,61)(49,53)(50,52)(51,55,59,57)(58,60);(30,32)(31,36)(33,41)(34,43)(35,42)(37,44)(38,40)(39,45)(46,48)(47,51)(49,57)(50,56)(52,59)(53,61)(54,60)(55,58);(30,33,35,34)(31,37,38,39)(32,41,42,43)(36,44,40,45)(47,52)(51,59)(53,60)(54,61)

Conjugacy classes & character table

754 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂₉×#8489, C₂₉×#8490, C₂₉×#8500, …
maximal normal subgroup of	C₂₉×#8489, C₂₉×#8490, C₂₉×#8500, …
maximal quotient of	C₂₉×#7594, C₂₉×#7596, C₂₉×#7599, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 267);