C₂₉×#1998 #16815712

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×#1998

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

228 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 112 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) = 53
transitive degree	μ_t(G) = 1856

A minimal faithful permutation representation — 3 generators on 53 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)(32,36)(33,34)(35,37)(38,39,42,45,41,44,46,52)(40,43,48,51,47,50,49,53);(31,34)(35,36)(38,40,46,48,41,47,42,49)(39,43,45,53,44,50,52,51);(30,32,33,37)(31,35,34,36)(39,44)(43,50)(45,52)(51,53)

Conjugacy classes & character table

928 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂₉×#8485, C₂₉×#8486, C₂₉×#10497, …
maximal normal subgroup of	C₂₉×#8485, C₂₉×#8486, C₂₉×#10497, …
maximal quotient of	C₂₉×#9880, C₂₉×#9881, C₂₉×#9892, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 730);