C₂₉×#1291 #16815005

order	3712
nilpotency class	3
isoclinism	isoclinic class of #1284
automorphism	?
homology	Abelianization (H₁) = C₄²×C₂₉ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₂₉×#1291

Subgroups

	subgroup	G/H
center, socle	C₂²×C₂₉	#95
commutator	C₂×C₄	C₄²×C₂₉
Frattini, cosocle	C₂×C₄²	C₂²×C₂₉
prime radical	C₂³×C₂₉	C₄⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₉ ◁ C₂×C₄²×C₂₉ ◁ G

138 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 138 conjugacy classes of subgroups — too many to draw. Only its 64 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 — 2 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)(32,34)(33,36)(35,37)(38,39,41,46,43,45,42,47)(40,48,49,53,50,44,51,52);(31,34,36,35)(38,40,42,49,43,50,41,51)(39,44,47,52,45,48,46,53)

Conjugacy classes & character table

928 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×C₂₉×#1291, C₂₉×#7992, C₂₉×#7993, …
maximal normal subgroup of	C₂×C₂₉×#1291, C₂₉×#7992, C₂₉×#7993, …
maximal quotient of	C₂₉×#7019, C₂₉×#7021, C₂₉×#7022, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 23);