C₂₉×#1470 #16815184

order	3712
nilpotency class	3
isoclinism	isoclinic class of #1468
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₂₉ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₂₉×#1470

Subgroups

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

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

normal subgroups only

This group has 204 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)(32,38,40,35)(33,36)(34,37)(39,44,45,42)(41,43)(46,47,49,52,50,53,56,59)(48,51,55,58,54,57,60,61);(30,32,34,40)(31,35,37,38)(33,39,41,45)(36,42,43,44)(46,48)(47,51)(49,55)(50,54)(52,58)(53,57)(56,60)(59,61);(30,33,34,41)(31,36,37,43)(32,39,40,45)(35,42,38,44)(48,54)(51,57)(55,60)(58,61)

Conjugacy classes & character table

928 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂₉×#8476, C₂₉×#8478, C₂₉×#8485, …
maximal normal subgroup of	C₂₉×#8476, C₂₉×#8478, C₂₉×#8485, …
maximal quotient of	C₂₉×#7657, C₂₉×#7660, C₂₉×#7663, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 202);