C₂₉×#1574 #16815288

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

Aliases

direct product

C₂₉×#1574

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×C₄⋊C₄
prime radical	C₂³×C₂₉	C₄⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₉ ◁ C₂×C₄²×C₂₉ ◁ G

158 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 158 conjugacy classes of subgroups — too many to draw. Only its 96 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)(30,31,35,37)(32,33,36,34)(38,39,41,44,42,45,48,51)(40,43,47,50,46,49,52,53);(30,32,34,31,35,36,33,37)(38,40)(39,43)(41,47)(42,46)(44,50)(45,49)(48,52)(51,53);(30,33,35,34)(31,36,37,32)(40,46)(43,49)(47,52)(50,53)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂₉×#8898, C₂₉×#8899, C₂₉×#8900, …
maximal normal subgroup of	C₂₉×#8898, C₂₉×#8899, C₂₉×#8900, …
maximal quotient of	C₂₉×#7880, C₂₉×#8279, C₂₉×#8600, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 306);