C₂₉×#1735 #16815449

order	3712
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	?
homology	Abelianization (H₁) = C₂×C₄²×C₂₉ · Schur multiplier (H₂) = C₂³×C₄

Aliases

direct product

C₂₉×#1735

Subgroups

	subgroup	G/H
center	C₂²×C₄×C₂₉	D₄
commutator	C₄	C₂×C₄²×C₂₉
Frattini, cosocle	C₂²×C₄	C₂³×C₂₉
socle	C₂³×C₂₉	C₂×D₄
prime radical	C₂⁴×C₂₉	D₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₂₉ ◁ C₂³×C₄×C₂₉ ◁ G

424 conjugacy classes of subgroups; only the 232 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 232 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)(32,33)(34,35,36,37)(38,39)(40,48)(41,45)(42,46)(43,47)(44,49)(50,53)(51,52);(31,33)(38,40,42,50,43,51,41,49)(39,44,45,52,47,53,46,48);(38,41,43,42)(39,45,47,46)(40,49,51,50)(44,52,53,48)

Conjugacy classes & character table

1624 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂₉×#7260, C₂₉×#7262, C₂₉×#10706, …
maximal normal subgroup of	C₂₉×#7260, C₂₉×#7262, C₂₉×#10706, …
maximal quotient of	C₂₉×#7872, C₂₉×#7875, C₂₉×#7878, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 467);