C₂₉×#1488 #16815202

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

Aliases

direct product

C₂₉×#1488

Subgroups

	subgroup	G/H
center	C₂×C₄×C₂₉	C₂×D₄
commutator	C₂×C₄	C₂²×C₄×C₂₉
Frattini, cosocle	C₄²	C₂³×C₂₉
socle	C₂²×C₂₉	C₂×C₂²⋊C₄
prime radical	C₂×#41×C₂₉	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₉ ◁ C₂×C₄²×C₂₉ ◁ G

246 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 100 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)(30,31,32,35,33,34,36,37)(38,39)(40,47)(41,45)(42,50)(43,46)(44,48)(49,52)(51,53);(38,40)(39,44)(41,48)(42,51)(43,49)(45,52)(46,53)(47,50);(31,34)(35,37)(38,41,43,42)(39,45,46,50)(40,48,49,51)(44,52,53,47)

Conjugacy classes & character table

1102 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂₉×#8412, C₂₉×#8413, C₂₉×#8414, …
maximal normal subgroup of	C₂₉×#8412, C₂₉×#8413, C₂₉×#8414, …
maximal quotient of	C₂₉×#7608, C₂₉×#7611, C₂₉×#7612, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 220);